Chapter 1: Mechanics

A machinist is using a wrench to loosen a nut. The wrench is 25.0 cm long and, he exerts a 17.0-N force at the end of the handle at $\mathbf{37}\mathbf{°}$with the handle (Fig. E10.7). (a) What torque does the machinist exerts about the center of the nut? (b) What is the maximum torque he could exert with this force, and how should the force be oriented?

Figure E10.7

C The coordinates of a bird flying in the -plane are given by $\mathit{x}\left(t\right)\mathbf{=}\mathit{\alpha }\mathit{t}$and $\mathit{y}\left(t\right)\mathbf{=}\mathbf{3}\mathbf{}\mathit{m}\mathbf{-}\mathit{\beta }{\mathit{t}}^{\mathbf{2}}$, where $\mathit{\alpha }\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{4}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$and$\mathit{\beta }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ .(a) Sketch the path of the bird between$\mathit{t}\mathbf{=}\mathbf{0}\mathbf{}\mathit{s}$ and $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{}\mathit{s}$. (b) Calculate the velocity and acceleration vectors of the bird as functions of time. (c) Calculate the magnitude and direction of the bird’s velocity and acceleration at $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{}\mathit{s}$. (d) Sketch the velocity and acceleration vectors at$\mathit{t}\mathbf{=}\mathbf{2}\mathbf{}\mathit{s}$ . At this instant, is the bird’s speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

Find the tension in each cord in Fig. E5.7 if the weight of the suspended object is w.

Force of a Golf Swing. A 0.0450-kg golf ball initially at rest is given a speed of 25 m/s when a club strikes it. If the club and ball are in contact for 2.00 ms, what average force acts on the ball? Is the effect of the ball’s weight during the time of contact significant? Why or why not?

A 40.0-N force stretches a vertical spring 0.250 m. (a) What mass must be suspended from the spring so that the system will oscillate with a period of 1.00 s? (b) If the amplitude of the motion is 0.050 m and the period is that specified in part (a), where is the object and in what direction is it moving 0.35 s after it has passed the equilibrium position, moving downward? (c) What force (magnitude and direction) does the spring exert on the object when it is 0.030 m below the equilibrium position, moving upward?

In Fig. P9.80, the cylinder and pulley turn without frictionabout stationary horizontal axles that pass through their centers.A light rope is wrapped around the cylinder, passes over thepulley, and has a 3.00-kg box suspended from its free end. Thereis no slipping between the rope and the pulley surface. The uniformcylinder has mass 5.00 kg and radius 40.0 cm. The pulley isa uniform disk with mass 2.00 kg and radius 20.0 cm. The box isreleased from rest and descends as the rope unwraps from thecylinder. Find the speed of the box when it has fallen 2.50 m.

A cube is placed so that one corner is at the origin and three edges are along the x-, y-, and z-axes of a coordinate system (Fig. P1.80). Use vectors to compute (a) the angle between the edge along the z-axis (line ab) and the diagonal from the origin to the opposite corner (line ad), and (b) the angle between line ac (the diagonal of a face) and line ad.

A large turntable with radius 6.00 m rotates about a fixed vertical axis, making one revolution in 8.00 s. The moment of inertia of the turntable about this axis is $\mathbf{1200}\mathbf{}\mathbf{kg}\mathbf{·}{\mathbf{m}}^{\mathbf{2}}$. You stand, barefooted, at the rim of the turntable and very slowly walk toward the center, along a radial line painted on the surface of the turntable. Your mass is 70.0 kg. Since the radius of the turntable is large, it is a good approximation to treat yourself as a point mass. Assume that you can maintain your balance by adjusting the positions of your feet. You find that you can reach a point 3.00 m from the center of the turntable before your feet begin to slip. What is the coefficient of static friction between the bottoms of your feet and the surface of the turntable?

A 0.100-kg stone rests on a frictionless, horizontal surface. A bullet of mass 6.00 g, traveling horizontally at 350 m/s, strikes the stone and rebounds horizontally at right angles to its original direction with a speed of 250 m/s. (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

Pyramid Builders. Ancient pyramid builders are balancing a uniform rectangular slab of stone tipped at an angle$\mathit{\theta }$above the horizontal using a rope (Given figure). The rope is held by five workers who share the force equally.

1. If $\mathit{\theta }\mathbf{=}{\mathbf{20.0}}^{\mathbf{\circ }}$, what force does each worker exert on the rope?
2. As $\mathit{\theta }$increases, does each worker have to exert more or less force than in part (a), assuming they do not change the angle of the rope? Why?
3. At what angle do the workers need to exert no force to balance the slab? What happens if$\mathit{\theta }$exceeds this value?