Chapter 1: Mechanics

Consider the system shown in Fig. P6.81. The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of kinetic friction between the 8.00-kg block and the table top is ${\mathit{\mu }}_{\mathbf{k}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{250}$. The blocks are released from rest. Use energy methods to calculate the speed of the 6.00-kg block after it has descended 1.50 m.

A hammer is hanging by a light rope from the ceiling of a bus. The ceiling is parallel to the roadway. The bus is traveling in a straight line on a horizontal street. You observe that the hammer hangs at rest with respect to the bus when the angle between the rope and the ceiling of the bus is $\mathbf{56}\mathbf{.}\mathbf{0}\mathbf{°}$. What is the acceleration of the bus?

As planets with a wide variety of properties are being discovered

outside our solar system, astrobiologists are considering whether and how life

could evolve on planets that might be very different from earth. One recently

discovered extrasolar planet, or exoplanet, orbits a star whose mass is 0.70

times the mass of our sun. This planet has been found to have 2.3 times the

earth’s diameter and 7.9 times the earth’s mass. For planets in this size range,

computer models indicate a relationship between the planet’s density and

composition:

Based on these data, what is the most likely composition of this planet? (a)

Mostly iron; (b) iron and rock; (c) iron and rock with some lighter elements; (d)

hydrogen and helium gases.

A local ice hockey team has asked you to design an apparatus for measuring the speed of the hockey puck after a slap shot. Your design is $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{m}$a -long, uniform rod pivoted about one end so that it is free to rotate horizontally on the ice without friction. The $\mathbf{0}\mathbf{.}\mathbf{800}\mathbf{kg}$rod has a light basket at the other end to catch the$\mathbf{0}\mathbf{.}\mathbf{163}\mathbf{kg}$puck. The puck slides across the ice with velocity $\overrightarrow{\mathbf{v}}$(perpendicular to the rod), hits the basket, and is caught. After the collision, the rod rotates. If the rod makes one revolution every $\mathbf{0}\mathbf{.}\mathbf{736}\mathbf{s}$after the puck is caught, what was the puck’s speed just before it hit the rod?

Hooke’s Law for a Wire. A wire of length${\mathit{l}}_{\mathit{0}}$and cross-sectional areasupports a hanging weight W.

1. Show that if the wire obeys Eq. (11.7), it behaves like a spring of force constant$\frac{\mathbf{A}\mathbf{Y}}{{\mathbf{l}}_{\mathbf{0}}}$, where Yis Young’s modulus for the wire material.
2. What would the force constant be for alength of 16-gauge (diameter = 1.291 mm) copper wire? See Table 11.1.
3. What wouldhave to be to stretch the wire in part (b) by 1.25 mm?

The Kinetic Energy of Running. Using Problem 9.81 as a guide, apply it to a person running at $\mathit{v}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathit{k}\mathit{m}\mathbf{/}\mathit{h}$ , with his arms and legs each swinging through $\mathbf{\pm }{\mathbf{30}}^{\mathbf{o}}$in $\frac{\mathbf{1}}{\mathbf{2}}\mathit{s}$ . As before, assume that the arms and legs are kept straight.

A ball is thrown straight up from the ground with speed ${\mathbf{v}}_{\mathbf{0}}$. At the same instant, a second ball is dropped from rest from a height H, directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which the two balls collide. (b) Find the value of H in terms of${\mathbf{v}}_{\mathbf{0}}$ and g such that at the instant when the balls collide, the first ball is at the highest point of its motion.

Two identical masses are released from rest in a smooth hemispherical bowl of radius R from the positions shown in Fig. P8.82. Ignore friction between the masses and the surface of the bowl. If the masses stick together when they collide, how high above the bottom of the bowl will they go after colliding?

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.

A segment of DNA is put in place and stretched. Given figure shows a graph of the force exerted on the DNA as a function of the displacement of the stage. Based on this graph, which statement is the best interpretation of the DNA’s behaviour over this range of displacements? The DNA

1. Does not follow Hooke’s law, because its force constant increases as the force on it increases
2. Follows Hooke’s law and has a force constant of about\(0.1{\rm{ pN/nm}}\)
3. Follows Hooke’s law and has a force constant of about\(10{\rm{ pN/nm}}\)
4. Does not follow Hooke’s law, because its force constant decreases as the force on it increase

The experiment is designed so that the seeds move no more than 0.20 mm between photographic frames. What minimum frame rate for the high-speed camera is needed to achieve this?

(a) 250 frames/s;

(b) 2500 frames/s;

(c) 25,000 frames/s;

(d) 250,000 frames/s.