Chapter 1: Mechanics

In your job as a mechanical engineer you are designing a flywheel and clutch-plate system like the one in Example 10.11. Disk A is made of a lighter material than disk B, and the moment of inertia of disk A about the shaft is one-third that of disk B. The moment of inertia of the shaft is negligible. With the clutch disconnected, A is brought up to an angular speed v0; B is initially at rest. The accelerating torque is then removed from A, and A is coupled to B. (Ignore bearing friction.) The design specifications allow for a maximum of 2400 J of thermal energy to be developed when the connection is made. What can be the maximum value of the original kinetic energy of disk A so as not to exceed the maximum allowed value of the thermal energy?

An object is moving along the x-axis. At t = 0 it has velocity ${\mathbf{v}}_{\mathbf{0}}\mathbf{x}\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{/}\mathbf{s}$. Starting at time t = 0 it has acceleration ${\mathbf{a}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{Ct}$, where C has units of $\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{3}}$. (a) What is the value of C if the object stops in 8.00 s after t = 0? (b) For the value of C calculated in part (a), how far does the object travel during the 8.00 s?

CP A 12.0kgmass, fastened to the end of an aluminium wire with an unstretched length of, is whirled in a vertical circle with a constant angular speed of. The cross-sectional area of the wire is 120rev/min. Calculate the elongation of the wire when the mass is

1. at the lowest point of the path and
2. at the highest point of its path.

A movie stuntman (mass 80.0 kg) stands on a window ledge 5.0 m above the floor (Fig. P8.81). Grabbing a rope attached to a chandelier, he swings down to grapple with the movie’s villain (mass 70.0 kg), who is standing directly under the chandelier. (Assume that the stuntman’s center of mass moves downward 5.0 m. He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is , how far do they slide?

Mass Mis distributed uniformly over a disk of radius a. Find the

gravitational force (magnitude and direction) between this disk-shaped mass

and a particle with mass mlocated a distance xabove the centre of the disk

(Fig. P13.81). Does your result reduce to the correct expression as xbecomes

very large? (Hint:Divide the disk into infinitesimally thin concentric rings, use

the expression derived in Exercise 13.35 for the gravitational force due to each

ring, and integrate to find the total force.)

Don’t Miss the Boat. While on a visit to Minnesota (“Land of 10,000 Lakes”), you sign up to take an excursion around one of the larger lakes. Whe\({\bf{1500}} - {\bf{kg}}\)n you go to the dock where the boat is tied, you find that the boat is bobbing up and down in the waves, executing simple harmonic motion with amplitude \({\bf{20}}\,{\bf{cm}}\). The boat takes \({\bf{3}}.{\bf{5}}\,{\bf{s}}\) to make one complete up-and-down cycle. When the boat is at its highest point, its deck is at the same height as the stationary dock. As you watch the boat bob up and down, you (mass \({\bf{60}}\,{\bf{kg}}\)) begin to feel a bit woozy, due in part to the previous night’s dinner of lutefisk. As a result, you refuse to board the boat unless the level of the boat’s deck is within \({\bf{10}}\,{\bf{cm}}\) of the dock level. How much time do you have to board the boat comfortably during each cycle of up-and-down motion.

A DNA molecule, with its double- helix structure, can in some situations behave like a spring. Measuring the force required to stretch single DNA molecules under various conditions can provide information about the biophysical properties of DNA. A technique for measuring the stretching force makes use of a very small cantilever, which consists of a beam that is supported at one end and is free to move at the other end, like a tiny diving board. The cantilever is constructed so that it obeys Hooke’s law—that is, the displacement of its free end is proportional to the force applied to it. Because different cantilevers have different force constants, the cantilever’s response must first be calibrated by applying a known force and determining the resulting deflection of the cantilever. Then one end of a DNA molecule is attached to the free end of the cantilever, and the other end of the DNA molecule is attached to a small stage that can be moved away from the cantilever, stretching the DNA. The stretched DNA pulls on the cantilever, deflecting the end of the cantilever very slightly. The measured deflection is then used to determine the force on the DNA molecule.

During the calibration process, the cantilever is observed to deflect by\(0.10{\rm{ nm}}\)when a force of\(3.0{\rm{ pN}}\)is applied to it. What deflection of the cantilever would correspond to a force of\(6.0{\rm{ pN}}\)? (a)\(0.07{\rm{ nm}}\); (b)\(0.14{\rm{ nm}}\); (c)\(0.20{\rm{ nm}}\); (d)\(0.40{\rm{ nm}}\).

You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a$\mathbf{0}.\mathbf{600}\mathbf{kg}$ ball that is traveling horizontally at $\mathbf{10}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$. Your mass is $70.0kg$. (a) If you catch the ball, with what speed do you and the ball move afterward? (b) If the ball hits you and bounces off your chest, so afterward it is moving horizontally at $\mathbf{8}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$ in the opposite direction, what is your speed after the collision?

An interesting, though highly impractical example of oscillation is the motion of an object dropped down a hole that extends from one side of the earth, through its center, to the other side. With the assumption (not realistic) that the earth is a sphere of uniform density, prove that the motion is simple harmonic and find the period. [Note: The gravitational force on the object as a function of the object’s distance rfrom the center of the earth was derived in Example 13.10 (Section 13.6). The motion is simple harmonic if the acceleration \({a_x}\) and the displacement from equilibrium x are related by Eq. (14.8), and the period is then\(T = \frac{{2\pi }}{\omega }\)].

In 1993 the radius of Hurricane Emily was about 350 km. The wind speed near the center (“eye”) of the hurricane, whose radius was about 30 km, reached about 200 km/h. As air swirled in from the rim of the hurricane toward the eye, its angular momentum remained roughly constant. Estimate (a) the wind speed at the rim of the hurricane; (b) the pressure difference at the earth’s surface between the eye and the rim. (Hint: See Table 12.1.) Where is the pressure greater? (c) If the kinetic energy of the swirling air in the eye could be converted completely to gravitational potential energy, how high would the air go? (d) In fact, the air in the eye is lifted to heights of several kilometers. How can you reconcile this with your answer to part (c)?