Chapter 1: Mechanics

You are designing an elevator for a hospital. The force exerted on a passenger by the floor of the elevator is not to exceed1.60times the passenger’s weight. The elevator accelerates upward with constant acceleration for a distance of3.0 mand then starts to slow down. What is the maximum speed of the elevator?

A cylindrical bucket, open at the top, is 25.0 cm high and 10.0 cm in diameter. A circular hole with a cross-sectional area 1.50 cm² is cut in the center of the bottom of the bucket. Water flows into the bucket from a tube above it at the rate of 2.40 x 10^-4m³/s. How high will the water in the bucket rise?

A physics professor is pushed up a ramp inclined upward at 30.0° above the horizontal as she sits in her desk chair, which slides on frictionless rollers. The combined mass of the professor and chair is 85.0 kg. She is pushed 2.50 m along the incline by a group of students who together exert a constant horizontal force of 600 N. The professor’s speed at the bottom of the ramp is 2.00 m/s. Use the work–energy theorem to find her speed at the top of the ramp.

Tidal Forces near a Black Hole.An astronaut inside a spacecraft,

which protects her from harmful radiation, is orbiting a black hole at a distance

of 120 km from its centre. The black hole is 5.00 times the mass of the sun and

has aSchwarzschild radius of 15.0 km. The astronaut is positioned inside the

spaceship such that one of her 0.030-kg ears is 6.0 cm farther from the black

hole than the centre of mass of the spacecraft and the other ear is 6.0 cm

closer. (a) What is the tension between her ears? Would the astronaut find it

difficult to keep from being torn apart by the gravitational forces? (Since her

whole body orbits with the same angular velocity, one ear is moving too slowly

for the radius of its orbit and the other is moving too fast. Hence her head

must exert forces on her ears to keep them in their orbits.) (b) Is the centre of

gravity of her head at the same point as the centre of

mass? Explain.

Cliff Height. You are climbing in the High Sierra when you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top; 8.00 s later you hear the sound of the rock hitting the ground at the foot of the cliff. (a) If you ignore air resistance, how high is the cliff if the speed of sound is 330 m/s? (b) Suppose you had ignored the time it takes the sound to reach you. In that case, would you have overestimated or underestimated the height of the cliff? Explain.

:A 2004 Prius with a 150-lb driver and no passengers weighs 3071 lb. The car is initially at rest. Starting at t = 0, a net horizontal force${\mathit{F}}_{\mathbf{x}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ in the +x-direction is applied to the car. The force as a function of time is given in Fig. P8.100. (a) For the time interval t = 0 to t = 4.50 s, what is the impulse applied to the car? (b) What is the speed of the car at t = 4.50 s? (c) At t = 4.50 s, the 3500-N net force is replaced by a constant net braking force . Once the braking force is first applied, how long does it take the car to stop? (d) How much work must be done on the car by the braking force to stop the car? (e) What distance does the car travel from the time the braking force is first applied until the car stops?

:BIOThe Kinetic Energy of Walking. If a person of mass Msimply moved forward with speed V, his kinetic energy would be $\frac{1}{2}M{V}^2$ . However, in addition to possessing a forward motion, various parts of his body (such as the arms and legs) undergo rotation. Therefore, his total kinetic energy is the sum of the energy from his forward motion plus the rotational kinetic

energy of his arms and legs. The purpose of this problem is to see how much this rotational motion contributes to the person’s kinetic energy. Biomedical measurements show that the arms and hands together typically make up 13% of a person’s mass, while the legs and feet together account for 37%. For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. In a brisk walk, the arms and legs each move through an angle of about $\pm {30}^0$ (a total of${60}^0$ ) from the vertical in approximately 1 second. Assume that they are held straight, rather than being bent, which is not quite true. Consider a 75-kg person walking at $5.0km/h$, having

arms 70 cm long and legs 90 cm long. (a) What is the average angular velocity of his arms and legs? (b) Using the average angular velocity from part (a), calculate the amount of rotational kinetic energy in this person’s arms and legs as he walks. (c) What is the total kinetic energy due to both his forward motion and his rotation? (d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

Consider the system shown in Fig. P6.81. The rope and pulley have negligible mass, and the pulley is frictionless. Initially the block is moving downward and the block is moving to the right, both with a speed of . The blocks come to rest after moving 2.00 m. Use the work–energy theorem to calculate the coefficient of kinetic friction between the 8.00-kg block and the table top.

You are standing on a bathroom scale in an elevator in a tall building. Your mass is 64 kg . The elevator starts from rest and travels upward with a speed that varies with time according to$\mathbf{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\left(3.0m/s^2\right)\mathbf{t}\mathbf{+}\left(0.20m/s^3\right){\mathbf{t}}^{\mathbf{2}}$ . When t = 4.0 s , what is the reading on the bathroom scale?

Water flows steadily from an open tank as in Fig. P12.81. The elevation of point 1 is 10.0 m, and the elevation of points 2 and 3 is 2.00 m. The cross-sectional area at point 2 is 0.0480 m²; at point 3 it is 0.0160 m². The area of the tank is very large compared with the cross-sectional area of the pipe. Assuming that Bernoulli’s equation applies, compute (a) the discharge rate in cubic meters per second and (b) the gauge pressure at point 2.