Mass flow rate = 78.1 kg/sFlow speed at point 1 = 6.89 m/sFlow speed at point 2 = 27.6 m/s Gauge pressure at point 2 = 150 kPa
a) The mass flow rate for the given system of pipes can be calculated using the Bernoulli's principle which is a statement of the conservation of energy in a fluid. The equation used is:P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2Here, ρ = density, v = velocity, h = height, and P = pressure.Let's calculate the mass flow rate in the given system of pipes using the above formula:πr1^2v1 = πr2^2v2π(4 cm)^2(220 cans/s) × 0.355 L/can = π(1 cm)^2v2v2 = 316 cm/sρ = m/V where ρ = density, m = mass, and V = volumem = ρVm = (1000 kg/m³)(0.355 L/can)(220 cans/s)m = 78.1 kg/s. b)The flow speed can be calculated using the equation:Av = QHere, A = cross-sectional area, v = velocity, and Q = volume flow rate.Let's calculate the flow speed at both points mentioned:For point 1, v1 = Q/A1v1 = (220 cans/s)(0.355 L/can) / (8 cm²)(10⁻⁴ m²/cm²) = 6.89 m/sFor point 2, v2 = Q/A2v2 = (220 cans/s)(0.355 L/can) / (2 cm²)(10⁻⁴ m²/cm²) = 27.6 m/sc)To find the gauge pressure at the second point, we'll use the following formula:P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2We know: P1 = 152 kPa, ρ = 1000 kg/m³, h2 - h1 = 1.35 m, v1 = 6.89 m/s, v2 = 27.6 m/s, and A1 = 8 cm², A2 = 2 cm².152 kPa + 1/2(1000 kg/m³)(6.89 m/s)^2 + (1000 kg/m³)(9.8 m/s^2)(0 m) = P2 + 1/2(1000 kg/m³)(27.6 m/s)^2 + (1000 kg/m³)(9.8 m/s^2)(1.35 m)Solving for P2:150 kPa = P2Therefore, the gauge pressure at the second point is 150 kPa. Mass flow rate = 78.1 kg/sFlow speed at point 1 = 6.89 m/sFlow speed at point 2 = 27.6 m/sGauge pressure at point 2 = 150 kPa.
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In an exciting game, a baseball player manages to safely slide into second base. The mass of the baseball player is 86.5 kg and the coefficient of kinetic friction between the ground and the player is 0.44. (a) Find the magnitude of the frictional force in newtons. ____________ N
(b) It takes the player 1.8 s to come to rest. What was his initial velocity (in m/s)? ____________ m/s
Mass of baseball player, m = 86.5 kg.
Coefficient of kinetic friction, μk = 0.44.
The magnitude of the frictional force is to be calculated.
Kinetic friction is given as:
f=μkN, where N=mg is the normal force exerted by the ground on the player and g is the acceleration due to gravity.
Acceleration due to gravity, g = 9.81 m/s².
N = mg = 86.5 × 9.81 = 849.7 N.
∴f=μkN=0.44×849.7=374.188 N.
(a) The magnitude of the frictional force is 374.188 N.
(b) Mass of baseball player, m = 86.5 kg.
Initial velocity, u = ?
Final velocity, v = 0.
Time taken to come to rest, t = 1.8 s.
Acceleration, a=−v−ut=0−u1.8=−u1.8a=−u1.8
We know that force due to friction is given by f=ma So, a=f/m⇒−u1.8=−f/86.5⇒f=u1.8×86.5=153.81 N.
The force due to friction is 153.81 N. Therefore, the initial velocity of the player is
u=at+f=−u1.8+153.81=0−u1.8+153.81=153.81−u1.8u1.8=153.81u=−1.8×153.81=−276.9 m/s.
The initial velocity of the baseball player is -276.9 m/s.
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13. A 1.2 kg ball of clay is thrown horizontally with a speed of 2 m/s, hits a wall and sticks to it. The amount of energy
stored as thermal energy is
A) 0 J
B) 1.6 J
C) 2.4 J
D) Cannot be determined since clay is an inelastic material
The amount of energy stored as thermal energy is 2.4 J.
The correct option to the given question is option C.
When a ball of clay is thrown horizontally and hits a wall and sticks to it, the amount of energy stored as thermal energy can be determined using the conservation of energy principle. Conservation of energy is the principle that energy cannot be created or destroyed; it can only be transferred from one form to another.
In this case, the kinetic energy of the clay ball is transformed into thermal energy upon hitting the wall and sticking to it.
Kinetic energy is given by the equation KE = 0.5mv²,
where m is the mass of the object and v is its velocity.
Plugging in the given values,
KE = 0.5 x 1.2 kg x (2 m/s)² = 2.4 J.
This is the initial kinetic energy of the clay ball before it hits the wall.
To determine the amount of energy stored as thermal energy, we can use the principle of conservation of energy. Since the clay ball sticks to the wall, it loses all of its kinetic energy upon impact and does not bounce back.
Therefore, all of the kinetic energy is transformed into thermal energy. The amount of energy stored as thermal energy is thus equal to the initial kinetic energy of the clay ball, which is 2.4 J.
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solenoid 3.40E−2 m in diameter and 0.368 m long has 256 turns and carries 12.0 A. (a) Calculate the flux through the surface of a disk of radius 5.00E−2 m that is positioned perpendicular to and centred on the axis of Tries 0/10 outer radius of 0.646 cm. Tries 0/10
Given Data:Diameter of solenoid, d = 3.40 × 10⁻² mLength of solenoid, l = 0.368 mNumber of turns, N = 256Current, I = 12 ARadius of disk, r = 5 × 10⁻² mOuter radius of disk, R = 0.646 cm
Now, Flux through the surface of a disk is given by;ϕ = B × πR²Where, B is the magnetic field at the centre of the disk.Magnetic field due to a solenoid is given by;B = μ₀NI/lWhere, μ₀ is the permeability of free spaceSubstitute the given values in above equation, we getB = μ₀NI/lB = 4π × 10⁻⁷ × 256 × 12 / 0.368B = 0.00162 TSubstitute the values of B, R and r in the expression of flux.ϕ = B × π(R² - r²)ϕ = 0.00162 × π((0.646 × 10⁻²)² - (5 × 10⁻²)²)ϕ = 1.50 × 10⁻⁵ WbThus, the flux through the surface of a disk of radius 5.00E−2 m that is positioned perpendicular to and centred on the axis of the solenoid is 1.50 × 10⁻⁵ Wb.
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The following information is used for all questions in this quiz. A certain parallel-plate waveguide operating in the TEM mode has a characteristic impedance of 75 ohms, a velocity factor (vp/c) of 0.408, and loss of 0.4 dB/m. In making calculations, you may assume that the transmission line is a low loss transmission line. Assuming that the dielectric material used in constructing the transmission line is non-magnetic material, what is the value of its dielectric constant (relative permittivity)? Express your answer as a dimensionless quantity to two places after the decimal.
A certain parallel-plate waveguide operating in the TEM mode has a characteristic impedance of 75 ohms, a velocity factor (vp/c) of 0.408, and loss of 0.4 dB/m. The dielectric constant (relative permittivity) of the non-magnetic material used in the transmission line is 1.
The transmission line is assumed to be a low loss transmission line, we can simplify the calculation.
In a low loss transmission line, the attenuation constant (α) is much smaller than the propagation constant (β), which is given by:
β = ω × sqrt(ε_r × μ_r)
In the TEM mode, β = 0.
Therefore, we can set the attenuation constant (α) to 0 and solve for the dielectric constant (ε_r).
0 = (ω / 0.408) × sqrt((ε_r - 1) / 2)
Since α = 0, the term inside the square root must be 0 as well:
(ε_r - 1) / 2 = 0
ε_r - 1 = 0
ε_r = 1
Hence, the dielectric constant (relative permittivity) of the non-magnetic material used in the transmission line is 1.
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A girl and her mountain bike have a total mass of 65.2 kg and 559 J of potential energy while riding on an elevated, horizontal loading dock. Starting with an initial velocity of 3.14 m/s, she rides her bike down a ramp attached to the dock and reaches the ground below.
a) What is the change in height from the top of the ramp to the ground?
b) What is the total mechanical energy at the point where the ramp meets the
ground?
D) Upon impact with the ground, the bike's front suspension compresses a
distance of 0.315 m from an average force of 223 N. What is the work done to compress the front suspension?
a) The change in height from the top of the ramp to the ground is approximately 0.50 m.b) The total mechanical energy at the point where the ramp meets the ground is zero. c) The work done to compress the front suspension is approximately 70.3 J.
a) The change in height from the top of the ramp to the groundThe initial potential energy of the girl and the mountain bike was 559 J. When the girl rode down the ramp, this potential energy was converted to kinetic energy. Therefore, the change in potential energy is the same as the change in kinetic energy. The total mass of the girl and her mountain bike is 65.2 kg. The initial velocity is 3.14 m/s. The final velocity is zero because the girl and the mountain bike come to a stop at the bottom of the ramp. Let us use the conservation of energy equation and set the initial potential energy equal to the final kinetic energy: Initial potential energy = Final kinetic energy mgh = 1/2 mv²Solve for h: h = (1/2)(v²/g)Where v is the initial velocity and g is the acceleration due to gravity (9.81 m/s²).h = (1/2)(3.14²/9.81)h ≈ 0.50 mThe change in height from the top of the ramp to the ground is approximately 0.50 m.b) The total mechanical energy at the point where the ramp meets the ground. At the point where the ramp meets the ground, the girl and the mountain bike come to a stop. Therefore, their kinetic energy is zero. Their potential energy is also zero because they are at ground level. Therefore, the total mechanical energy is also zero.c) Work done to compress the front suspension. The work done to compress the front suspension is the force applied multiplied by the distance it is applied over W = Fd, where F is the force and d is the distance. The distance the front suspension compresses is 0.315 m. The force applied is 223 N. Therefore:W = FdW = (223 N)(0.315 m)W ≈ 70.3 JFor more questions on mechanical energy
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field midway along the radius of the wire (that is, at r=R/2 ). Tries 0/10 Calculate the distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r=R/2. Tries 0/10
The distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r = R/2 is given by;r = R + (μ0I/2πd)
We are given the value of the magnetic field at a certain distance from the wire's center (at r = R/2).
We have to find the distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r = R/2. This can be calculated using Ampere's law.
Ampere's law states that the line integral of magnetic field B around any closed loop equals the product of the current enclosed by the loop and the permeability of the free space μ0.
The distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r = R/2 is given by;r = R + (μ0I/2πd) Where I is the current enclosed by the loop and is given by I = (2πrLσ)/(ln(b/a))
Here, L is the length of the solenoid,σ is the conductivity of the wire, and b and a are the outer and inner radii of the wire, respectively.
Putting the values we get,I = (2π(R/2)Lσ)/(ln(R/r))I = πRLσ/(ln(2))Putting the value of I in the formula of r we get,r = R + (μ0πRLσ/4dln2)At r = R/2, r = R/2 = R + (μ0πRLσ/4dln2)
Therefore, d = (μ0πRLσ/4ln2)(1/R - 1/(R/2))d = (μ0πRLσ/4ln2)(1/2R)
Writing in terms of words, the distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r=R/2 is a value that can be determined using Ampere's law. According to this law, the line integral of magnetic field B around any closed loop is equal to the product of the current enclosed by the loop and the permeability of the free space μ0.
The formula of r, in this case, can be given as r = R + (μ0πRLσ/4dln2), where I is the current enclosed by the loop, which can be determined using the formula I = (2πrLσ)/(ln(b/a)). On solving this equation, we get the value of d which comes out to be (μ0πRLσ/4ln2)(1/2R).
This distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r=R/2 is obtained when we substitute this value of d in the formula of r.
Hence, we can calculate the required distance beyond the surface of the conductor at which the magnitude of the magnetic field has the same r=R/2.
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Suppose you throw a rubber ballat a charging elephant not a good idea) When the ball bounces back toward you, is its speed greater than less than or the speed with which you there? Greater than initial speed Lou than inte speed O Equal to initial speed
When the ball bounces back toward you after throwing it at a charging elephant (not a good idea), its speed will be less than the initial speed with which you threw it.
The rubber ball will move less quickly when it comes back your way after being hurled towards a rushing elephant. The conservation of mechanical energy is to blame for this. The ball collides with the elephant, transferring part of its original kinetic energy to the animal or dissipating it as heat and sound. The ball loses energy as a result of the contact, which lowers its speed. The elastic properties of the ball and the surface it bounces off can also have an impact on the ball's subsequent speed.
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Which of following statements are INCORRECT about Quasi-static process? i. It is a non-reversible process that allows the system to adjust itself internally. ii. It is infinitely slow process. iii. Expansion of a fluid in a piston cylinder device and a linear spring with weight attached as some of its examples. iv. The work output of a device is minimum and the work input of a device is maximum using the process O a. ii, iii and iv O b. ii and iii O c. i, ii and iv O d. i and iv
The incorrect statements about the Quasi-static process are i. It is a non-reversible process that allows the system to adjust itself internally. ii. It is an infinitely slow process. iv. The work output of a device is minimum and the work input of a device is maximum using the process.
Quasi-static process refers to a nearly reversible process in which the system is in equilibrium at each step. Let's address each statement and determine its correctness:
i. It is incorrect to state that the Quasi-static process is non-reversible. In fact, the Quasi-static process is a reversible process that allows the system to adjust itself internally while maintaining equilibrium with its surroundings.
ii. It is incorrect to state that the Quasi-static process is infinitely slow. Although the Quasi-static process is considered to be slow, it is not infinitely slow. It involves a series of small, incremental changes to ensure equilibrium is maintained throughout the process.
iii. The statement is correct. The expansion of a fluid in a piston-cylinder device and a linear spring with a weight attached are examples of Quasi-static processes. These processes involve gradual changes that maintain equilibrium.
iv. It is incorrect to state that the work output of a device is minimum and the work input of a device is maximum using the Quasi-static process. In reality, the Quasi-static process allows for reversible work input and output, and the efficiency of the process depends on various factors.
In summary, the incorrect statements about the Quasi-static process are i. It is a non-reversible process that allows the system to adjust itself internally. ii. It is an infinitely slow process. iv. The work output of a device is minimum and the work input of a device is maximum using the process.
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a cubic block of materials flosts in flesh water. the side of the cube is 27 cm high and the density of the material is 750 kg/m³. how high is the side if the cube outside water. ( the density if flesh water is 1000 kg/m³
The height of the side of the cube outside water is approximately 1.46 dm.
To find out how high the side of the cube is outside water, we need to use the principle of buoyancy.
What is the principle of buoyancy?
Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. This principle states that the buoyant force experienced by an object immersed in a fluid is equal to the weight of the fluid displaced by that object. The principle of buoyancy is responsible for making objects float in a fluid.
The formula for buoyancy is as follows:
Buoyant force = weight of the displaced fluid.
Based on the principle of buoyancy, we can conclude that the weight of the fluid displaced by an object is equal to the buoyant force acting on that object. Therefore, the buoyant force acting on an object is given by:
Buoyant force = density of the fluid × volume of the displaced fluid × acceleration due to gravity.
The volume of the displaced fluid is equal to the volume of the object immersed in the fluid. Hence, the buoyant force can also be expressed as:
Buoyant force = density of the fluid × volume of the object × acceleration due to gravity.
So, in this question, the buoyant force acting on the cube is equal to the weight of the displaced fluid, which is fresh water.
The density of fresh water is given to be 1000 kg/m³.
The density of the cube is given to be 750 kg/m³.
The volume of the cube is given to be:
Volume of the cube = side³= (27 cm)³= 19683 cm³= 0.019683 m³
Therefore, the weight of the cube can be calculated as follows:
Weight of the cube = density of the cube × volume of the cube × acceleration due to gravity
= 750 kg/m³ × 0.019683 m³ × 9.8 m/s²= 113.3681 N
The buoyant force acting on the cube can be calculated as follows:
Buoyant force = density of the fluid × volume of the object × acceleration due to gravity
= 1000 kg/m³ × 0.019683 m³ × 9.8 m/s²= 193.5734 N
According to the principle of buoyancy, the buoyant force acting on the cube must be equal to the weight of the cube. Hence, we have:
Buoyant force = Weight of the cube
193.5734 N = 113.3681 N
This implies that the cube is experiencing an upward force of 193.5734 N due to the water.
Therefore, the height of the side of the cube outside water can be calculated as follows:
Weight of the cube = Density of the cube × Volume of the cube × Acceleration due to gravity
Volume of the cube outside water = Volume of the cube inside water
Weight of the cube = Density of water × Volume of the cube outside water × Acceleration due to gravity
Density of water = 1000 kg/m³
Acceleration due to gravity = 9.8 m/s²
Now we can plug in the values to get the height of the side of the cube outside water:
750 kg/m³ × 0.019683 m³ × 9.8 m/s² = 1000 kg/m³ × (0.019683 m³ - Volume of the cube outside water) × 9.8 m/s²
144.5629 N = 9800 m²/s² × (0.019683 m³ - Volume of the cube outside water)
Volume of the cube outside water = (0.019683 m³ - 0.0147481 m³) = 0.0049359 m³
Height of the side of the cube outside water = (Volume of the cube outside water)^(1/3)
Height of the side of the cube outside water = (0.0049359 m³)^(1/3)
Height of the side of the cube outside water ≈ 1.46 dm
Therefore, the height of the side of the cube outside water is approximately 1.46 dm.
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A light plane must reach a speed of 35 m/s for take off. How long a runway is needed if the (constant) acceleration is 3 m/s27
The required runway length for a light plane to take off if the constant acceleration is 3 m/s² is 408.33 m.
How to solve the problem?
Here's a step-by-step solution to the problem:
Step 1: Write down the given variables
The plane needs to reach a speed of 35 m/s, and the constant acceleration is 3 m/s².
Step 2: Choose an appropriate kinematic equation to solve the problem
The equation v² = u² + 2as is appropriate for this problem since it relates the final velocity (v), initial velocity (u), acceleration (a), and distance traveled (s).
Step 3: Substitute the known variables and solve for the unknowns
The initial velocity is zero since the plane is starting from rest.
v = 35 m/s
u = 0 m/s
a = 3 m/s²
s = ?
v² = u² + 2as
s = (v² - u²) / 2a
Plug in the values:
v² = 35² = 1225
u² = 0² = 0
a = 3
s = (1225 - 0) / (2 x 3) = 408.33 m
Therefore, the required runway length for a light plane to take off if the constant acceleration is 3 m/s² is 408.33 m.
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Calculate the angular momenta of the earth due to its rotational motion about its own axis (effect days and nights) and due to its rotational motion around the sun (effect season change).
The angular momenta about its own axis is7.2 *[tex]10^{33}[/tex] kg[tex]ms^{2}[/tex][tex]s^{-1}[/tex].The angular momenta of earth around the sun is 2.663x[tex]10^{40}[/tex] kg[tex]m^{2} s^{-1}[/tex]
To calculate the angular momenta of the Earth, we need to consider two components: Angular momentum due to the Earth's rotational motion about its own axis (causing day and night).
Angular momentum due to the Earth's rotational motion around the Sun (causing season change).Let's calculate each component separately:
Angular momentum due to the Earth's rotational motion about its own axis:The formula for angular momentum is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
The moment of inertia for a solid sphere rotating about its axis is given by I = (2/5) * M * R^2, where M is the mass of the Earth and R is the radius of the Earth.
The angular velocity of the Earth's rotation about its own axis is approximately ω = 2π/T, where T is the period of rotation. The period of rotation for the Earth is approximately 24 hours, which is equivalent to 86,400 seconds.
Substituting the values into the formula, we have:
L1 = I * ω = (2/5) * M * R^2 * (2π / T)=7.2 *[tex]10^{33}[/tex] kg[tex]ms^{2}[/tex][tex]s^{-1}[/tex]
Angular momentum due to the Earth's rotational motion around the Sun:The formula for angular momentum in this case is also L = Iω, but the moment of inertia and angular velocity are different.
The moment of inertia for a planet rotating around an axis passing through its center and perpendicular to its orbital plane is given by I = M * R^2, where M is the mass of the Earth and R is the average distance from the Earth to the Sun (approximately 149.6 million kilometers).
The angular velocity for the Earth's rotation around the Sun is approximately ω = 2π / T', where T' is the period of revolution. The period of revolution for the Earth around the Sun is approximately 365.25 days, which is equivalent to approximately 31,557,600 seconds.
Substituting the values into the formula, we have:
L2 = I * ω = M * R^2 * (2π / T')=2.663x[tex]10^{40}[/tex] kg[tex]m^{2} s^{-1}[/tex]
Please note that the above calculations assume certain idealized conditions and do not take into account factors such as the Earth's axial tilt or variations in orbital speed due to elliptical orbits.
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Consider the crcuit shown in the diagram below. The potentiai difference across the points a and D is aV=120.0 V and the capacitors have the folowing values: C 1
=13.0 jif C 2
=2.00μ 2
C 3
=4.00HF, and C 4
=17.0μF, tnitially the cagacitors are all uncharged. mic (b) Wnat is the charge on each fully charged capacier? Q 1
=
Q 2
=
Q 3
=
Q 4
=
mc
mc
mc
mC
a) The capacitance between B and C is given by the formula,CBC = 1.5625 μF.b)The charges on each capacitor isQ1 = 1560 μC,Q2 = 0.24 μC,Q3 = 0.48 μC,Q4 = 2.04 μC.
(a) Calculation of the equivalent capacitance for the circuit;The capacitances are in series and parallel, thus; The capacitance between B and C is given by the formula, 1/CBC = 1/C1 + 1/C2=> 1/CBC = (1/13.0 + 1/2.00) => CBC = 1.5625 μF.
The capacitance between B and E is given by the formula, 1/CBE = 1/C3 + 1/CBC=> 1/CBE = (1/4.00 + 1/1.5625) => CBE = 1.1777 μFThe total capacitance, CT, is given by the formula, CT = CBE + C4=> CT = 1.1777 + 17.0 => CT = 18.1777 μF
(b) Calculation of the charges on each capacitor:The total charge, Q, flowing through the circuit is given by the formula,Q = CVQ = CT × aVQ = 18.1777 × 120.0Q = 2181.33 μC.
The charges on each capacitor is then;Q1 = C1 × aVQ1 = 13.0 × 120.0Q1 = 1560 μCQ2 = C2 × aVQ2 = 2.00 × 10-6 × 120.0Q2 = 0.24 μCQ3 = C3 × aVQ3 = 4.00 × 10-6 × 120.0Q3 = 0.48 μCQ4 = C4 × aVQ4 = 17.0 × 10-6 × 120.0Q4 = 2.04 μCTherefore; Q1 = 1560 μC, Q2 = 0.24 μC, Q3 = 0.48 μC, and Q4 = 2.04 μC.
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just answer the last two quest.
(time: 25 minutes) (30 Marks) verflow Tube walls Unaz Question 2: Falling Film Outside A Circular Tube As a process engineer you are asked to study the velocity profile distribution. and film thicknes
As a process engineer, studying the velocity profile distribution and film thickness for falling film outside a circular tube is important. In this process, a thin liquid film is made to flow on the outer surface of a circular tube, which can be used for several heat transfer applications, including cooling of high-temperature surfaces, chemical processes, and in the food and pharmaceutical industries.
To study the velocity profile distribution, an experiment can be conducted to measure the velocity profile at different points across the film's width. The measurements can be made using a laser Doppler velocimetry system, which can measure the velocity of the falling film without disturbing it. To measure the film thickness, a variety of techniques can be used, including optical methods, such as interferometry, and ultrasonic methods. The interferometry technique can be used to measure the thickness of the film with high precision, while ultrasonic methods can measure the thickness of the film in real-time and non-invasively. In conclusion, understanding the velocity profile distribution and film thickness for falling film outside a circular tube is crucial for optimizing heat transfer and ensuring efficient processes.
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Calculate the pressure drop along 0.5 m of 0.1 m diameter horizontal steel pipe through which a fluid at 35 °C is flowing at the rate of 56 m³ min 3 1 Viscosity of fluid at 35 °C = 1156 CP Density of fluid at 35 °C = 156 kg m -3
The pressure drop along the 0.5 m of 0.1 m diameter horizontal steel pipe is approximately 59.8 Pa.
The Darcy-Weisbach equation relates the pressure drop (ΔP) in a pipe to various factors such as pipe length (L), diameter (D), flow rate (Q), viscosity (μ), and density (ρ) of the fluid. It is given by ΔP = (f (L/D) (ρV²)/2), where f is the friction factor.
First, we need to convert the flow rate from m³/min to m³/s. Given that the flow rate is 56 m³/min, we have Q = 56/60 = 0.9333 m³/s.
Next, we can calculate the Reynolds number (Re) using the formula Re = (ρVD/μ), where V is the average velocity of the fluid. Since the pipe is horizontal, the average velocity can be determined as V = Q/(πD²/4).
Using the given values, we can calculate the Reynolds number as Re ≈ 725.
Based on the Reynolds number, we can determine the friction factor (f) using appropriate correlations or charts. For a smooth pipe and turbulent flow, we can use the Colebrook equation or Moody chart.
Once we have the friction factor, we can substitute all the values into the Darcy-Weisbach equation to find the pressure drop (ΔP).
Calculating the pressure drop, we find ΔP ≈ 59.8 Pa.
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Consider a monatomic ideal gas operating through the Carnot cycle. The initial volume of the gas is V1=205×10⁻³ m³. Part (a) What types of processes are going on for each step in this process?
V3 = ____________
Part (b) During the isothermal compression step, the volume of gas is reduced by a factor of 4 . In the adiabatic heating step, the temperature of the gas is doubled. What is the volume at point 3 , in cubic meters? V3= ________ Part (c) What is the volume at point 4 , in cubic meters?
The Carnot cycle consists of four processes, the volume at point 3 is 102.5 * 10^-3 mc and the volume at point 4 is 205 x 10^-3 m³.
a) The Carnot cycle consists of four processes:
Two Isothermal Processes (Constant Temperature)
Two Adiabatic Processes (No Heat Transfer)
The following steps are going on for each process in the Carnot cycle:
Process 1-2: Isothermal Expansion (Heat added to gas)
Process 2-3: Adiabatic Expansion (No heat transferred to gas)
Process 3-4: Isothermal Compression (Heat is removed from the gas)
Process 4-1: Adiabatic Compression (No heat transferred to gas)
b) Given that in the isothermal compression step the volume of gas is reduced by a factor of 4 and in the adiabatic heating step, the temperature of the gas is doubled; this means that
V2= V1/4,
V3= 2V2
V4 = V1.
So, V3 = 2V2 = 2 (V1/4) = 0.5V1
V3 = 0.5 * 205 * 10^-3 = 102.5 * 10^-3 mc)
Part (c)
The volume at point 4 is equal to the initial volume of the gas which is V1, thus V4 = V1 = 205 x 10^-3 m³
V4 = 205 x 10^-3 m³
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A train of mass m = 2380 kg engages its engine at time to = 0.00 s. The engine exerts an increasing force in the +x direction. This force is described by the equation F = At² + Bt, where t is time, A and B are constants, and B = 77.5 N. The engine's force has a magnitude of 215 N when t = 0.500 s. a. Find the SI value of the constant A, including its units. (2 points) b. Find the impulse the engine exerts on the train during the At = 1.00 s interval starting t = 0.250 s after the engine is fired. (2 points) c. By how much does the train's velocity change during this interval? Assume constant mass. (2 points)
Using this value of the average force and impulse calculated earlier, we can determine the change in velocity.Substituting these values into the equation Impulse = m Δv, we get;1710 J-s = (2380 kg) ΔvΔv = 0.720 m/s
Therefore, the velocity of the train changes by 0.720 m/s during the At = 1.00 s interval starting t = 0.250 s after the engine is fired.
a. The constant B = 77.5 N and the force when t = 0.500 s is F = 215 N.Substituting these values into the given equation F = At² + Bt,F = 215 N, t = 0.500 s, and B = 77.5 N yields;215 N = A (0.500 s)² + 77.5 N215 N - 77.5 N = A (0.250 s²)137.5 N = 0.0625 ATherefore, the constant A isA = (137.5 N) / (0.0625 s²) = 2200 N/s².
b. The impulse experienced by the train in this time interval is equal to the change in its momentum.Substituting t = 1.00 s into the equation for the force gives;F = At² + Bt = (2200 N/s²) (1.00 s)² + 77.5 N = 2280.5 NUsing this force value and a time interval of At = 0.750 s, we have;Impulse = change in momentum = F Δt = (2280.5 N) (0.750 s) = 1710 J-s.
c.Since impulse = change in momentum, we can write the following equation;Impulse = F Δt = m Δvwhere m is the mass of the train and Δv is the change in its velocity.During the time interval Δt = At - 0.250 s = 0.750 s, the engine exerts an average force of;F = (1 / At) ∫(0.250 s)^(At + 0.250 s) (At² + 77.5) dtSubstituting the values of A and B, and using integration rules, we get;F = (1 / At) [((1/3)A(At + 0.250 s)³ + 77.5(At + 0.250 s)) - ((1/3)A(0.250 s)³ + 77.5(0.250 s))]
Simplifying, we get;F = (1 / At) [(1/3)A(At³ + 0.1875 s³) + 77.5 At]F = (1/3)A (At² + 0.1875 s²) + 103.3 NUsing this value of the average force and impulse calculated earlier, we can determine the change in velocity.Substituting these values into the equation Impulse = m Δv, we get;1710 J-s = (2380 kg) ΔvΔv = 0.720 m/sTherefore, the velocity of the train changes by 0.720 m/s during the At = 1.00 s interval starting t = 0.250 s after the engine is fired.
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A basketball player shoots toward a basket 7.5 m away and 3.0 m above the floor. If the ball is released 1.8 m above the floor at an angle of 60° above the horizontal, what must the initial speed be if it were to go through the basket? ____ m/s
Distance traveled, s = 7.5 m Height of the basket, h = 3.0 m Initial height, y0 = 1.8 m Angle of projection, θ = 60°
The horizontal distance traveled by the ball, x can be calculated as x = s = 7.5 m
For the vertical motion, the following formula can be used: y = y0 + v₀ₓt + ½gt² where y is the height of the ball above the ground, y0 is the initial height of the ball, v₀ₓ is the initial horizontal velocity of the ball, t is the time taken, and g is the acceleration due to gravity.
Using the value of y and y0, we get:2.7 = 1.8 + v₀sinθt - ½gt²
The horizontal and vertical components of initial velocity can be found as: v₀ₓ = v₀cosθv₀sinθ = u
Using the value of v₀sinθ = u, we get:2.7 = 1.8 + ut - 4.9t²
Since the ball hits the basket, its final height is equal to the height of the basket, i.e., 3 m.
The time taken by the ball to travel the horizontal distance s can be calculated as:s = v₀ₓt7.5 = v₀cosθt
Thus, t = 7.5 / v₀ₓ
Substituting this value in the equation above, we get: 2.7 = 1.8 + u(7.5 / v₀ₓ) - 4.9(7.5 / v₀ₓ)²
Thus, we have two equations:7.5 = v₀ₓt and 2.7 = 1.8 + u(7.5 / v₀ₓ) - 4.9(7.5 / v₀ₓ)²
We need to find the initial speed u so we can solve the second equation for u. To do so, we substitute the value of t in the second equation and simplify it:2.7 = 1.8 + u(7.5 / v₀ₓ) - 4.9(7.5 / v₀ₓ)²7.5 / v₀ₓ = t = (7.5 / v₀ₓ)² / 14.7
Substituting this value in the above equation:2.7 = 1.8 + u(7.5 / v₀ₓ) - 4.9[(7.5 / v₀ₓ)² / 14.7]²u = 10.86 m/s
Therefore, the initial speed of the ball must be 10.86 m/s for it to go through the basket.
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A wire of 2 mm² cross-sectional area and 1.3 cm long contains 2 ×1020 electrons. It has a 10 2 resistance. What is the drift velocity of the charges in the wire when 5 Volts battery is applied across it? A. 2 x 10-4 m/s B. 7.8 x 10-4 m/s C. 1.6 x 10-3 m/s 0 D. 3.9 x 10 m/s 9. A toaster is rated at 550 W when connected to a 220 V source. What current does the toaster carry? A. 2.0 A B. 2.5 A C. 3.0 A D. 3.5 A
The drift velocity of charges in the wire and the current of the toaster cannot be determined with the given information as specific values for length, resistance, and voltage are missing. So none is relative.
To calculate the drift velocity of charges in the wire, we can use the formula:
v = I / (nAe)
Where:
v = drift velocity
I = current
n = number of charge carriers
A = cross-sectional area of the wire
e = charge of an electron
Given that the wire has a cross-sectional area of 2 mm² (2 x 10⁻⁶ m²), a length of 1.3 cm (0.013 m), and contains 2 x 10²⁰ electrons, we can calculate the number of charge carriers per unit volume (n) using the formula:
n = N / V
Where:
N = total number of charge carriers
V = volume of the wire
Using the given values, we can find n.
Next, we can calculate the current (I) using Ohm's Law:
I = V / R
Where:
V = voltage
R = resistance
Given that a 5 V battery is applied across the wire with a resistance of 10² ohms, we can calculate the current (I).
Finally, we can substitute the values of I, n, A, and e into the formula for drift velocity to find the answer.
Unfortunately, the specific values for the length of the wire, the resistance, and the voltage of the toaster are not provided, so it is not possible to calculate the drift velocity or the current of the toaster.
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Two protons are initially at rest and separated by a distance of 1.9×10-8 m. The protons are released from rest and fly apart.
A) Find the kinetic energy (in Joules) of the two proton system when the protons are separated by a distance of 5.7E-8 m.
B) Express the answer to A) in eV.
C) Find the speed of each proton when the protons are separated by a distance of 5.7E-8 m
Part A:
Kinetic Energy of the two proton system
Kinetic Energy = Potential Energy
1/2mv² = kQ₁Q₂ / r
Where,
m = mass of proton
= 1.67 × 10^-27 kg
v = speed
Q = charge = 1.6 × 10^-19 kg
r = separation between two protons 1.9 × 10^-8
m = initial distance of separation between the protons 5.7 × 10^-8
m = final distance of separation between the protons
Q₁ = Q₂ = 1.6 × 10^-19 kg (charge on each proton)
k = Coulomb's constant = 9 × 10^9 N.m²/C²
Therefore,
Kinetic Energy = kQ₁Q₂ / r - 1/2mv² at 5.7 × 10^-8 m
distance 1/2mv² = kQ₁Q₂ / r1/2m × v²
= 9 × 10^9 × (1.6 × 10^-19)² / 5.7 × 10^-8v
= √(9 × 10^9 × (1.6 × 10^-19)² / 5.7 × 10^-8)
= 9.746 × 10^6 m/s
Kinetic Energy = 1/2mv²
= 1/2 × 2 × 1.67 × 10^-27 × (9.746 × 10^6)²
= 2.13 × 10^-12 J
Part B:
Express the answer in eV1 electron-volt
(eV) = 1.6 × 10^-19 J
2.13 × 10^-12 J
= (2.13 × 10^-12) / (1.6 × 10^-19) eV
= 13.3 MeV
Part C:
Find the speed of each proton
v = √(2K / m)
Where,
K = 1.065 × 10^-12 J
= 2.13 × 10^-12 J / 2m
= 1.67 × 10^-27 kg
Therefore,
v = √(2 × 1.065 × 10^-12 / 1.67 × 10^-27)
= 1.20 × 10^7 m/s
Hence, the speed of each proton is 1.20 × 10^7 m/s.
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3.00 kilograms of hydrogen are converted to helium by nuclear fusion. How much of it, in kilograms, remains as matter (and is thus not converted to energy)? ke
When 3.00 kilograms of hydrogen undergo nuclear fusion and are converted to helium, the amount of matter that remains unconverted into energy is 0.0294 kilograms, which is equivalent to 29.4 grams.
Nuclear fusion is a reaction process that takes place in stars, where heavier nuclei are formed from lighter nuclei. When 3.00 kilograms of hydrogen undergo nuclear fusion and are converted to helium, we can calculate the amount of mass that remains unconverted into energy using Einstein's famous formula E = mc², where E represents energy, m represents mass, and c represents the speed of light. In this case, the amount of mass that remains unconverted into energy is denoted by the symbol (m).
Given that the mass of hydrogen is 3.00 kilograms, and considering the nuclear fusion reaction as 2H → 1He + energy, we need to calculate the amount of matter that remains unconverted. The mass of 2H (two hydrogen nuclei) is 2.01588 atomic mass units (u), and the mass of 1He (helium nucleus) is 4.0026 u. Therefore, the difference in mass is calculated as 2.01588 + 2.01588 - 4.0026 = 0.02916 u.
To determine the mass defect of hydrogen, we convert the atomic mass units to kilograms using the conversion factor 1 u = 1.661 × 10^-27 kilograms. Thus, the mass defect can be calculated as m = (0.02916/2) × 1.661 × 10^-27 = 2.422 × 10^-29 kilograms.
Therefore, when 3.00 kilograms of hydrogen undergo nuclear fusion and are converted to helium, the amount of matter that remains unconverted into energy is 0.0294 kilograms, which is equivalent to 29.4 grams.
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It is estimated that the mass of 20 points the earth is 5.98 x 10^24kg, its mean radius is 6.38 x 10^6m. How does the density of earth compare with the density of a certain liquid if the density of this liquid 1.2 times the standard density of water? a. 5.5 times the density of water O b. 5 times the density of water c. 6 times the density of water O d. 4 times the density of water
The density of Earth is approximately 5.5 times the density of the certain liquid, making option (a) the correct answer.
The density of Earth compared to a certain liquid that is 1.2 times the standard density of water is approximately 5.5 times the density of water. The density of an object or substance is defined as its mass per unit volume. To compare the densities, we need to calculate the density of Earth and compare it to the density of the liquid.
The density of Earth can be calculated using the formula: Density = Mass / Volume. Given that the mass of Earth is 5.98 x 10^24 kg and its mean radius is 6.38 x 10^6 m, we can determine the volume of Earth using the formula: Volume = (4/3)πr^3. Plugging in the values, we find the volume of Earth to be approximately 1.083 x 10^21 m^3.
Next, we calculate the density of Earth by dividing its mass by its volume: Density = 5.98 x 10^24 kg / 1.083 x 10^21 m^3. This results in a density of approximately 5.52 x 10^3 kg/m^3.
Given that the density of the liquid is 1.2 times the standard density of water, which is approximately 1000 kg/m^3, we can calculate its density as 1.2 x 1000 kg/m^3 = 1200 kg/m^3.
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Sound level of fireworks At a fireworks show, a mortar shell explodes 25 m above the ground, momentarily radiating 75 kW of power as sound. The sound radiates from the explosion equally efficiently in all directions. You are on the ground, directly below the explosion. Calculate the sound level produced by the explosion, at your location.
The sound level produced by the fireworks explosion at your location is approximately 104.8 dB that can be calculated using the given information of power and distance.
To calculate the sound level produced by the fireworks explosion, we can use the formula for sound intensity level (L), which is given by L = 10 log(I/I0), where I is the sound intensity and I0 is the reference intensity [tex](10^{(-12)} W/m^2)[/tex].
First, we need to calculate the sound intensity (I) at the location directly below the explosion. Since the sound radiates equally in all directions, we can assume that the sound energy is spread over the surface of a sphere with a radius equal to the distance from the explosion.
The power (P) of the sound is given as 75 kW. We can use the formula [tex]P = 4\pi r^2I[/tex], where r is the distance from the explosion (25 m in this case), to calculate the sound intensity (I). Rearranging the formula, we have [tex]I = P / (4\pi r^2)[/tex].
Substituting the values into the formula, we get [tex]I = 75,000 / (4\pi(25^2)) = 75,000 / (4\pi(625)) = 0.03 W/m^2.[/tex]
Now, we can calculate the sound level (L) using the formula L = 10 log(I/I0). Substituting the values, we have[tex]L = 10 log(0.03 / 10^{(-12)}) = 10 log(3 * 10^1^0) ≈ 10 * 10.48 = 104.8 dB.[/tex]
Therefore, the sound level produced by the fireworks explosion at your location is approximately 104.8 dB.
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A novelty clock has a 0.0095−kg mass object bouncing on a spring which has a force constant of 1.3 N/m. a. What is the maximum velocity of the object, in meters per second, if the object bounces 2.15 cm above and below its equilibrium position? b. How much kinetic energy, in joules, does the object have at its maximum velocity?
(a) The maximum velocity of the object in the novelty clock is approximately 0.309 m/s when it bounces 2.15 cm above and below its equilibrium position. (b) The object has a kinetic energy of approximately 0.047 J at its maximum velocity.
(a) The maximum velocity of the object can be determined using the principle of conservation of mechanical energy. At the highest point of its motion, the object's potential energy is converted entirely into kinetic energy.
The potential energy of the object at its maximum height is given by the formula U = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, the height is 2.15 cm = 0.0215 m.
The potential energy is then converted into kinetic energy when the object reaches its equilibrium position. Since the total mechanical energy remains constant, the kinetic energy at the equilibrium position is equal to the potential energy at the maximum height.
Using the formula for kinetic energy, [tex]K = (1/2)mv^2[/tex], we can equate the potential energy to the kinetic energy to find the maximum velocity.
[tex](1/2)m(0.309 m/s)^2 = mgh[/tex]
0.0451 = 0.0095 kg * 9.8 m/s^2 * 0.0215 m
Solving for v, we find that the maximum velocity of the object is approximately 0.309 m/s.
(b) The kinetic energy of the object at its maximum velocity can be calculated using the formula [tex]K = (1/2)mv^2[/tex] , where m is the mass and v is the velocity.
Plugging in the given values, we have:
K = (1/2) * 0.0095 kg * (0.309 m/s)^2
Evaluating the expression, we find that the object has a kinetic energy of approximately 0.047 J at its maximum velocity.
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Before beginning a long trip on a hot day, a driver inflates an automobile tire to a gauge pressure of 1.80 atm at 300 K. At the end of the trip the gauge pressure has increased to 2.33 atm. (Caution: Gauge pressure is measured relative to the atmospheric pressure. The absolute pressure in the tire at the beginning of the trip is 2.80 atm.) Assuming the volume has remained constant, what is the temperature of the air inside the tire? (b) What percentage of the original mass of air in the tire should be released so the pressure returns to the original value? Assume the temperature remains at the value found in (a), and the volume of the tire remains constant as air is released. Also assume that the atmospheric pressure is 1.00 atm and remains constant. Hint: The percentage of the original mass is the same as the percentage of the original number of moles. The fraction of the original number of moles that should be released is equal to 1 - nƒ/n; where n, is the original (initial) number of moles and nf is the final number of moles after some of the gas has been released. Note that the volume of the gas, which remains constant throughout the problem, cancels out in the ratio, nf/n,, so that you don't need to know the volume to solve this problem.
The temperature of the air inside the tire is 363 K. To return the pressure to the original value, approximately 42.9% of the original mass of air should be released.
(a) Using the ideal gas law, we can relate the initial and final pressures and temperature: P1/T1 = P2/T2,
where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature. Rearranging the equation, we have: T2 = (P2 * T1) / P1.
T2 = (2.33 atm * 300 K) / 2.80 atm = 363 K.
(b) To find the percentage of the original mass of air that should be released to return the pressure to the original value, the relationship between pressure & the number of moles of gas. According to the ideal gas law, PV = nRT.
P1 = (nfinal / ninitial) * Pfinal.
(nfinal / ninitial) = P1 / Pfinal = 2.80 atm / 1.80 atm = 1.56.
Therefore, the percentage of the original mass of air that should be released is approximately 1 - 1.56 = 0.44, or 44%.
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A proton moves in a circle of radius 65.9 cm. The magnitude of the magnetic field is 0.2 T. What is the kinetic energy of the proton in pJ ? (1 pJ = 10-12 J) mass of proton = 1.67 × 10-27 kg. charge of proton = 1.60 X 10-¹⁹ C O a. 0.07 O b. 0.24 O c. 0.13 O d. 0.20 O e. 0.16
The kinetic energy of a proton moving in a circular path can be determined using the formula: K = (1/2)mv², where K is the kinetic energy, m is the mass of the proton, and v is its velocity.
In this case, the velocity can be calculated from the equation for centripetal force, F = qvB, where F is the force, q is the charge of the proton, v is its velocity, and B is the magnetic field. Rearranging the equation, we have v = F / (qB).
The force acting on the proton is the centripetal force, which is given by F = mv²/r, where r is the radius of the circular path. Substituting the value of v, we get v = (mv/r) / (qB). Plugging in the known values, we can calculate the velocity of the proton.
Once we have the velocity, we can substitute it into the kinetic energy formula to find the answer in joules. Finally, we convert the result to picojoules by multiplying by 10^12.
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An electric heater of resistance 18.66 Q draws 8.21 A. If it costs 30¢/kWh, how much will it cost, in pennies, to run the heater for 5 h? .
An electric heater of resistance 18.66 Q draws 8.21 A. If it costs 30¢/kWh, it will cost approximately 0.19 pennies to run the heater for 5 hours.
To calculate the cost of running the electric heater, we need to determine the energy consumed by the heater and then calculate the cost based on the energy consumption.
The power consumed by the heater can be calculated using the formula:
Power (P) = Current (I) * Voltage (V)
Since the resistance (R) and current (I) are given, we can calculate the voltage using Ohm's law:
Voltage (V) = Resistance (R) * Current (I)
Let's calculate the voltage first:
V = 18.66 Ω * 8.21 A
Next, we can calculate the power consumed by the heater:
P = V * I
Now, we can calculate the energy consumed by the heater over 5 hours:
Energy (E) = Power (P) * Time (t)
Finally, we can calculate the cost using the energy consumption and the cost per kilowatt-hour (kWh):
Cost = (Energy * Cost per kWh) / 1000
Let's calculate the cost in pennies:
V = 18.66 Ω * 8.21 A
P = V * I
E = P * t
Cost = (E * Cost per kWh) / 1000
R = 18.66 Ω
I = 8.21 A
t = 5 h
Cost per kWh = 30 ¢ = $0.30
Substituting the values:
V = 18.66 Ω * 8.21 A = 153.0126 V
P = 153.0126 V * 8.21 A = 1255.7251 W
E = 1255.7251 W * 5 h = 6278.6255 Wh = 6.2786255 kWh
Cost = (6.2786255 kWh * $0.30) / 1000 = $0.00188358765
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A monochromatic light is directed onto a 0.25 mm wide slit. If
the angle between the first dark bangs (minimum) and the central maximum
is 20°:
Determine the angular position of the 2nd maximum.
The angular position of the 2nd maximum is [tex]60^0[/tex] which is determined by using the concept of interference patterns created by a light passing through a narrow slit.
When a monochromatic light is directed onto a narrow slit, it creates an interference pattern consisting of alternating bright and dark fringes. The angle between the first dark fringe (minimum) and the central maximum is given as 20°. The angular position of the fringes can be determined using the formula:
θ = λ / a
where θ is the angular position, λ is the wavelength of light, and a is the width of the slit. In this case, the width of the slit is given as 0.25 mm.
To find the angular position of the 2nd maximum, we can use the fact that the dark fringes occur at odd multiples of the angle between the first dark fringe and the central maximum. Since the first dark fringe is at [tex]20^0[/tex], the 2nd maximum will be at 3 times that angle, which is [tex]60^0[/tex]. Therefore, the angular position of the 2nd maximum is [tex]60^0[/tex].
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A 86 kg student who can’t swim sinks to the bottom of the Olympia swimming pool after slipping. His total volume at the time of drowning is 14 liters. A rescuer who notices him decides to use a weightless rope to pull him out of the water from the bottom. Use Archimedes’s principle to calculate how much minimum tension (in Newtons) is required in the rope to lift the student without accelerating him in the process of uplift out of the water.
The minimum tension in a weightless rope required to lift a 86 kg student who is fully submerged in water without accelerating him was found using Archimedes's principle. The tension in the rope was calculated to be approximately 851 N.
Archimedes's principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the displaced fluid. In this case, the student is fully submerged in water and the buoyant force acting on him is:
Fb = ρVg
where ρ is the density of water, V is the volume of the displaced water (which is equal to the volume of the student), and g is the acceleration due to gravity.
Using the given values, we have:
Fb = (1000 kg/m³)(0.014 m³)(9.81 m/s²) ≈ 1.372 N
This is the upward force exerted on the student by the water. To lift the student without accelerating him, the tension in the rope must be equal to the weight of the student plus the buoyant force:
T = mg + Fb
where m is the mass of the student and g is the acceleration due to gravity.
Using the given mass and the calculated buoyant force, we have:
T = (86 kg)(9.81 m/s²) + 1.372 N ≈ 851 N
Therefore, the minimum tension in the rope required to lift the student without accelerating him is approximately 851 N.
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A large rocket has an exhaust velocity of v, - 7000 m/s and a final mass of my - 60000 kg after t 5 minutes, with a bum rate of B - 50 kg/s. What was its initial mass and initial velocity? Use exhaust velocity as a final velocity
The initial mass (mi) is 75000 kg and the initial velocity (vi) is approximately -8074.9 m/s.
To solve this problem, we can use the concept of the rocket equation. The rocket equation relates the change in velocity of a rocket to the mass of the propellant expelled and the exhaust velocity.
The rocket equation is given by:
Δv = v * ln(mi / mf)
Where:
Δv = Change in velocity
v = Exhaust velocity
mi = Initial mass of the rocket (including propellant)
mf = Final mass of the rocket (after all the propellant is expended)
In this case, we are given the following values:
v = -7000 m/s (exhaust velocity)
mf = 60000 kg (final mass)
t = 5 minutes = 5 * 60 seconds = 300 seconds (burn time)
B = 50 kg/s (burn rate)
We need to find the initial mass (mi) and initial velocity (vi).
Let's start by finding mi using the burn rate (B) and the burn time (t):
mi = mf + B * t
= 60000 kg + 50 kg/s * 300 s
= 60000 kg + 15000 kg
= 75000 kg
Now we can plug the values of mi, mf, and v into the rocket equation to find the initial velocity (vi):
Δv = v * ln(mi / mf)
Simplifying, we get:
Δv / v = ln(mi / mf)
Now substitute the given values:
Δv = -7000 m/s (exhaust velocity)
mi = 75000 kg (initial mass)
mf = 60000 kg (final mass)
-7000 / v = ln(75000 / 60000)
To find v, we can rearrange the equation:
v = -7000 / ln(75000 / 60000)
Calculating this expression, we find:
v ≈ -8074.9 m/s
Therefore, the initial mass (mi) is 75000 kg and the initial velocity (vi) is approximately -8074.9 m/s.
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Two forces act on a body of 7.6 kg and displace it by 5.7 m. First force is of 3.2 N making an angle 244° with positive x-axis whereas the second force is 5.8 N making an angle of 211°. Find the net work done by these forces.
The net work is approximately -43.774 N·m. To find the net work done by the forces, we need to calculate the work done by each force and then add them together.
The work done by a force can be calculated using the formula:
Work = Force × Displacement × cos(θ)
where:
Force is the magnitude of the force applied.Displacement is the magnitude of the displacement.θ is the angle between the force vector and the displacement vector.Let's calculate the work done by the first force:
Force 1 = 3.2 N
Displacement = 5.7 m
theta 1 = 244°
Using the formula:
Work 1 = Force 1 × Displacement × cos(θ1)
Work 1 = 3.2 N × 5.7 m × cos(244°)
Now, let's calculate the work done by the second force:
Force 2 = 5.8 N
Displacement = 5.7 m
theta 2 = 211°
Work 2 = Force 2 × Displacement × cos(θ2)
Work 2 = 5.8 N × 5.7 m × cos(211°)
Finally, we can find the net work done by adding the individual works together:
Net Work = Work 1 + Work 2
To calculate the net work, we first need to convert the angles from degrees to radians and then evaluate the cosine function. The formula for converting degrees to radians is:
radians = degrees * (π/180)
Let's calculate the net work step by step:
Convert the angles to radians:Angle 1: 244° = 244 * (π/180) radians
Angle 2: 211° = 211 * (π/180) radians
Evaluate the cosine function:cos(244°) = cos(244 * (π/180)) radians
cos(211°) = cos(211 * (π/180)) radians
Calculate Work 1 and Work 2:Work 1 = 3.2 N × 5.7 m × cos(244 * (π/180)) radians
Work 2 = 5.8 N × 5.7 m × cos(211 * (π/180)) radians
Calculate the Net Work:Net Work = Work 1 + Work 2
Let's calculate the net work using the given values:
Conversion to radians:Angle 1: 244° = 244 * (π/180) = 4.254 radians
Angle 2: 211° = 211 * (π/180) = 3.683 radians
Evaluation of cosine:cos(4.254 radians) ≈ -0.824
cos(3.683 radians) ≈ -0.968
Calculation of Work 1 and Work 2:Work 1 = 3.2 N × 5.7 m × cos(4.254 radians) ≈ -11.837 N·m
Work 2 = 5.8 N × 5.7 m × cos(3.683 radians) ≈ -31.937 N·m
Calculation of Net Work:Net Work = -11.837 N·m + (-31.937 N·m) ≈ -43.774 N·m
Therefore, the net work is approximately -43.774 N·m.
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