Chapter 1: Mechanics

At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye’s T3 law:

$\mathbf{C}\mathbf{=}\mathbf{K}\frac{{\mathbf{T}}^{\mathbf{3}}}{{\mathbf{\theta }}^{\mathbf{3}}}$

where $\mathbf{k}\mathbf{=}\mathbf{}\mathbf{1940}\mathbf{}\mathbf{J}\mathbf{/}\mathbf{mol}\mathbf{}\mathbf{x}\mathbf{}\mathbf{K}\mathbf{}\mathbf{and}\mathbf{}\mathbf{\theta }\mathbf{=}\mathbf{281}\mathbf{K}$. (a) How much heat is required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K? (Hint: Use Eq. (17.18) in the form dQ = nCdT and integrate.) (b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at 40.0 K?

A sphere with radius R= 0.200 m has density that decreases with distance rfrom the center of the sphere according to$\mathit{\rho }\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}\mathbf{-}\left(3.00\times {10}^{3}kg/m^4\right)\mathit{r}$ (a) Calculate the total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

A pump is required to lift 800 kg of water (about 210 gallons) per minute from a well 14.0 m deep and eject it with a speed of 18.0 m/s. (a) How much work is done per minute in lifting the water? (b) How much work is done in giving the water the kinetic energy it has when ejected? (c) What must be the power output of the pump?

The upper ball is released from rest, collides with the stationary lower ball, and sticks to it. The strings are both$\mathbf{50}\mathbf{}\mathbf{cm}$long. The upper ball has mass 2.00 kg, and it is initially 10.0 cm higher than the lower ball, which has mass 3.00 kg. Find the frequency and maximum angular displacement of the motion after the collision.

A 500.0-g bird is flying horizontally at 2.25 m/s, not paying much attention, when it suddenly flies into a stationary vertical bar, hitting it 25.0 cm below the top (Fig. P10.85). The bar is uniform, 0.750 m long, has a mass of 1.50 kg, and is hinged at its base. The collision stuns the bird so that it just drops to the ground afterward (but soon recovers to fly happily away). What is the angular velocity of the bar (a) just after it is hit by the bird and (b) just as it reaches the ground?

Question:Two identical balls, each in diameter, are suspended by two wires (Fig. P5.85). The entire apparatus is supported by a single wire, and the surfaces of the balls are perfectly smooth. (a) Find the tension in each of the three wires. (b) How hard does each ball push on the other one?

You are to use a long, thin wire to build a pendulum in a science museum. The wire has an unstretched length of 22.00mand a circular cross-section of diameter 0.860 mm; it is made of an alloy that has a large breaking stress. One end of the wire will be attached to the ceiling, and a 9.50 kgmetal sphere will be attached to the other end. As the pendulum swings back and forth, the wire’s maximum angular displacement from the vertical will be$\mathbf{36}\mathbf{.}\mathbf{0}\mathbf{°}$. You must determine the maximum amount the wire will stretch during this motion. So, before you attach the metal sphere, you suspend a test mass (mass m) from the wire’s lower end. You then measure the increase in lengthof the wire for several different test masses. Given figure, a graph ofversusshows the results and the straight line that gives the best fit to the data. The equation for this line is$\mathbf{∆}\mathbf{l}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{422}\mathbf{m}\mathbf{/}\mathbf{kg}\mathbf{)}\mathbf{m}$.

1. Assume that g = 9.8 m/localid="1668148375172" ${\mathbf{s}}^{\mathbf{2}}$, and use given figure to calculate Young’s modulusfor this wire.
2. You remove the test masses, attach the 9.50 kgsphere, and release the sphere from rest, with the wire displaced by 36.localid="1668148446283" $\mathbf{0}\mathbf{°}$. Calculate the amount the wire will stretch as it swings through the vertical. Ignore air resistance.

Question: A model car starts from rest and travels in a straight line. A smartphone mounted on the car has an app that transmits the magnitude of the car’s acceleration (measured by an accelerometer) every second. The results are given in the table:

Each measured value has some experimental error. (a) Plot acceleration versus time and find the equation for the straight line that gives the best fit to the data. (b) Use the equation for a (t) that you found in part (a) to calculate v(t) , the speed of the car as a function of time. Sketch the graph of versus . Is this graph a straight line? (c) Use your result from part (b) to calculate the speed of the car at t = 5.00s. (d) Calculate the distance the car travels between t = 0 and t = 5.00s.

Question:Traffic Court.You are called as an expert witness in a trial for a traffic violation. The facts are these: A driver slammed on his brakes and came to a stop with constant acceleration. Measurements of his tires and the skid marks on the pavement indicate that he locked his car’s wheels, the car traveled before stopping, and the coefficient of kinetic friction between the road and his tires was 0.750 . He was charged with speeding in a 45- mi/h zone but pleads innocent. What is your conclusion: guilty or innocent? How fast was he going when he hit his brakes?