Chapter 1: Mechanics

Neutron Stars and Supernova Remnants. The Crab Nebula is a cloud of glowing gas about 10 lightyears across, located about 6500 light years from the earth (Fig. P9.86). It is the remnant of a star that underwent a

supernova explosion,seen on earth in 1054 A.D. Energy is released by the Crab Nebula at a rate of about $\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{31}}\mathit{W}$, about ${\mathbf{10}}^{\mathbf{5}}$times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron starat its center. This object rotates once every 0.0331 s, and this period is increasing by$\mathbf{4}\mathbf{.}\mathbf{22}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{13}}\mathit{s}$ for each second of time that elapses. (a) If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star. (b) Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers. (c) What is the linear speed of a point on the equator of the neutron star? Compare to the speed of light. (d) Assume that the neutron star is uniform and calculate its

density. Compare to the density of ordinary rock ($\mathbf{3000}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$) and to the density of an atomic nucleus (about ${\mathbf{10}}^{\mathbf{7}}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$). Justify the statement that a neutron star is essentially a large atomic nucleus.

A small block with mass 0.130 kg is attached to a string passing through a hole in a frictionless, horizontal surface (see Fig. E10.40). The block is originally revolving in a circle with a radius of 0.800 m about the hole with a tangential speed of 4.00 m/s. The string is then pulled slowly from below, shortening the radius of the circle in which the block revolves. The breaking strength of the string is 30.0 N. What is the radius of the circle when the string breaks?

The Grand Coulee Dam is 1270 m long and 170 m high. The electrical power output from generators at its base is approximately 2000 MW. How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 92 % of the work done on the water by gravity is converted to electrical energy? (Each cubic meter of water has a mass of 1000 kg.)

A 5.00-g bullet is shot through a 1.00-kg wood block suspended on a string 2.00 m long. The center of mass of the block rises a distance of 0.38 cm. Find the speed of the bullet as it emerges from the block if its initial speed is 450 m/s.

Question: The Silently Ringing Bell. A large, 34.0-kg bell is hung from a wooden beam so it can swing back and forth with negligible friction. The bell’s center of mass is 0.60 m below the pivot. The bell’s moment of inertia about an axis at the pivot is \({\bf{18}}.{\bf{0}}{\rm{ }}{\bf{kg}} \cdot {{\bf{m}}^2}\). The clapper is a small, 1.8-kg mass attached to one end of a slender rod of length L and negligible mass. The other end of the rod is attached to the inside of the bell; the rod can swing freely about the same axis as the bell. What should be the length L of the clapper rod for the bell to ring silently—that is, for the period of oscillation for the bell to equal that of the clapper?

In the vertical jump, an athlete starts from a crouch and jumps upward as high as possible. Even the best athletes spend little more than 1.00 s in the air (their “hang time”). Treat the athlete as a particle and let${\mathit{Y}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$localid="1655800915664" ${\mathit{Y}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$be his maximum height above the floor. To explain why he seems to hang in the air, calculate the ratio of the time he is abovelocalid="1655800919441" ${\mathit{Y}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{/}\mathbf{2}$to the time it takes him to go from the floor to that height. Ignore air resistance.

A technician is testing a computer-controlled, variable-speed motor. She attaches a thin disk to the motor shaft, with the shaft at the center of the disk. The disk starts from rest, and sensors attached to the motor shaft measure the angular acceleration ${\mathit{\alpha }}_{\mathbf{z}}$of the shaft as a function of time. The results from one test run are shown in Fig. P9.87:

(a) Through how many revolutions has the disk turned in the first 5.0 s? Can you use Eq. (9.11)? Explain. What is the angular velocity, in rad/s, of the disk (b) at t= 5.0 s; (c) when it has turned through 2.00 rev?

A 55-kg runner runs around the edge of a horizontal turntable mounted on a vertical, frictionless axis through its center. The runner’s velocity relative to the earth has magnitude 2.8 m/s. The turntable is rotating in the opposite direction with an angular velocity of magnitude 0.20 rad/s relative to the earth. The radius of the turntable is 3.0 m, and its moment of inertia about the axis of rotation is80kgm². Find the final angular velocity of the system if the runner comes to rest relative to the turntable. (You can model the runner as a particle.)

A physics student spends part of her day walking between classes or for recreation, during which time she expends energy at an average rate of 280 W. The remainder of the day she is sitting in class, studying, or resting; during these activities, she expends energy at an average rate of 100 W. If she expends a total of $\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathbf{\hspace{0.33em}}\mathit{J}$ of energy in a 24 hour day, how much of the day did she spend walking?

You need to measure the mass Mof a 4.00 mlong bar. The bar has a square cross section but has some holes drilled along its length, so you suspect that its center of gravity isn’t in the middle of the bar. The bar is too long for you to weigh on your scale. So, first, you balance the bar on a knife-edge pivot and determine that the bar’s center of gravity is 1.88 mfrom its left-hand end. You then place the bar on the pivot so that the point of support is 1.50 mfrom the left-hand end of the bar. Next you suspend amass${\mathbf{m}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{kg}$from the bar at a point from the left-hand end. Finally, you suspend a massfrom the bar at a distance xfrom the left-hand end and adjustso that the bar is balanced. You repeat this step for other values ofand record each corresponding value of. The table gives your result

1. Draw a free-body diagram for the bar whenandare suspended from it.
2. Apply the static equilibrium equation$\mathbf{\sum }_{\mathbf{}}{\mathbf{\tau }}_{\mathbf{z}}\mathbf{=}\mathbf{0}$with the axis at the location of the knife-edge pivot. Solve the equation for xas a function of${\mathit{m}}_{\mathbf{2}}$.
3. Plot xversus$\frac{\mathbf{1}}{{\mathbf{m}}_{\mathbf{2}}}$. Use the slope of the best-fit straight line and the equation you derived in part (b) to calculate that bar’s mass M. Use$\mathbf{g}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{80}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$.
4. What is the y-intercept of the straight line that fits the data? Explain why it has this value