I have a few physics questions that I don't understand. Two are thermal expansion problems and the other is an ideal gas law problem.
1. Two rods, one made of iron and the other of aluminum, have the same initial length at 64.00 deg. C? When the temperature drops to 4.00 deg. C the aluminum rod is 1.900 mm shorter than the iron rod. What was the initial length of the rods?
The aluminum coefficient is 26x10^-6 and the iron rod coefficient is 12x10^-6.
2. A plate made of steel has a 18.9979 mm diameter hole cut into it. A 19.0000 mm diameter ball made of glass is sitting in the hole. Both are at an initial temperature of 9.00 deg. C. If the plate and ball are both heated, at what temperature will the ball just fit through the hole?
(Steel coefficient = 12x10^-6 and glass coefficient = 9x10^-6).
For the next problem, I got the first part but I can't get part B.
3. A. What is the gauge pressure in a 29.0 deg. C car tire with a volume of 39.00 litres and containing 2.20 moles of gas?
For my answer I got: 4.03x10^4 Pa
3B. If 4.00 litres of gas initially at 1.00 atm pressure and 29.0 deg. C is added to tire, what will the gauge pressure be?
How should I manipulate the equations for these problems? Thank you!!
1.
The aluminum coefficient is 26x10^-6 and the iron rod coefficient is 12x10^-6.
deltaL=L*coeff*(Tf-Ti)
aluminum
Liron-.0019=L*cal*(4-64)
iron:
Liron=L*cfe(4-64)
substuting from the second into the first.
L*cfe(-60)-.0019=L*cal*(-60)
you take if from here, solve for L
3. find the moles of added gas, then using ideal gas law, change moles by adding.
To solve these thermal expansion and ideal gas law problems, you need to use relevant equations and manipulate them to find the desired quantities. Let's go through each problem step by step:
1. Thermal Expansion Problem:
In this problem, we are given the initial and final temperatures, coefficients of thermal expansion, and the difference in length between the two rods.
The formula for linear expansion is given by:
ΔL = αLΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion
L is the initial length
ΔT is the change in temperature
To find the initial length of the rods, we need to set up equations for both aluminum and iron and solve them simultaneously.
For aluminum:
ΔL_aluminum = α_aluminum * L_aluminum * ΔT
For iron:
ΔL_iron = α_iron * L_iron * ΔT
From the problem statement, we have the following information:
ΔL_aluminum = -1.900 mm
ΔT = (4.00 - 64.00) = -60.00 °C
Plugging in the values and rearranging the equations, we get:
L_aluminum = (ΔL_aluminum) / (α_aluminum * ΔT)
L_iron = (ΔL_iron) / (α_iron * ΔT)
Now substitute the coefficients of thermal expansion:
L_aluminum = (-1.900 * 10^-3) / (26 * 10^-6 * (-60.00))
L_iron = (-1.900 * 10^-3) / (12 * 10^-6 * (-60.00))
Calculate both L_aluminum and L_iron to find the initial lengths of the rods.
2. Thermal Expansion Problem:
In this problem, we need to find the temperature at which the glass ball just fits through the hole in the steel plate.
We can use the same formula for linear expansion as in problem 1 to set up equations for both the glass ball and the steel plate.
For the steel plate:
ΔL_plate = α_plate * L_plate * ΔT
For the glass ball:
ΔL_ball = α_ball * L_ball * ΔT
Since we want the ball to just fit in the hole, the change in length of the plate is equal to the difference in radius between the hole and the diameter of the ball (assuming the hole's diameter remains constant).
ΔL_plate = (D_hole / 2 - D_ball / 2)
D_hole = 18.9979 mm
D_ball = 19.0000 mm
Now we can rewrite the equation as:
(D_hole - D_ball) / 2 = α_plate * L_plate * ΔT
Rearrange and solve for ΔT:
ΔT = [(D_hole - D_ball) / 2] / (α_plate * L_plate)
Substitute the coefficients and dimensions, and solve for ΔT.
3. Ideal Gas Law Problem:
For both part A and part B of this problem, we can use the ideal gas law equation to solve for the gauge pressure.
The ideal gas law equation is:
PV = nRT
Where:
P is the pressure
V is the volume
n is the number of moles
R is the ideal gas constant
T is the temperature
For part A, we are given the volume, number of moles, temperature, and need to find pressure.
Substitute the given values into the ideal gas law equation, and solve for pressure.
For part B, we are given an additional volume of gas to add to the tire. To find the new gauge pressure, we can see that the volume becomes V_initial + V_added, and the number of moles remains the same. Keep the initial pressure and temperature the same, and solve for the new pressure using the ideal gas law equation.
With these steps and equations, you should be able to manipulate them and find the answers to the given problems. Good luck!