Chapter 1: Mechanics

A thin, uniform rod is bent into a square of side length a.

If the total mass is M, find the moment of inertia about an axis

through the center and perpendicular to the plane of the square.

(Hint:Use the parallel axis theorem.)

CP An amusement park ride consists of airplane-shaped cars attached to steel rods (Given figure). Each rod has a length of 15.00 mand a cross-sectional area of 8.00${\mathbf{cm}}^{\mathbf{2}}$.

1. How much is each rod stretched when it is vertical and the ride is at rest? (Assume that each car plus two people seated in it has a total weight of 1900 N.)
2. When operating, the ride has a maximum angular speed of $\mathbf{12}\frac{\mathbf{rev}}{\mathbf{min}}$. How much is the rod stretched then?

A 20.00-kg lead sphere is hanging from a hook by a thin wire 2.80 m long and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00-kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

Observations of this planet over time show that it is in a nearly circular orbit around its star and completes one orbit in only 9.5 days. How many times the orbital radius rof the earth around our sun is this exoplanet’s orbital radius around its sun? Assume that the earth is also in a nearly circular orbit. (a) 0.026r; (b) 0.078r; (c) 0.70r; (d) 2.3r.

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.

The stage moves at a constant speed while stretching the DNA. Which of the graphs in Fig. given below best represents the power supplied to the stage versus time?

If a seed is launched at an angle of 0° with the maximum initial speed, how far from the plant will it land? Ignore air resistance, and assume that the ground is flat. (a) 20 cm; (b) 93 cm; (c) 2.2 m; (d) 4.6 m.

A 4.00-g bullet, traveling horizontally with a velocity of magnitude 400 m/s, is fired into a wooden block with mass 0.800 kg, initially at rest on a level surface. The bullet passes through the block and emerges with its speed reduced to 190 m/s. The block slides a distance of 72.0 cm along the surface fromits initial position. (a) What is the coefficient of kinetic friction between block and surface? (b) What is the decrease in kinetic energy of the bullet? (c) What is the kinetic energy of the block at the instant after the bullet passes through it?

Question: In a physics lab experiment, you release a small steel ball at various heights above the ground and measure the ball’s speed just before it strikes the ground. You plot your data on a graph that has the release height (in meters) on the vertical axis and the square of the final speed (in ) on the horizontal axis. In this graph your data points lie close to a straight line. (a) Using and ignoring the effect of air resistance, what is the numerical value of the slope of this straight line? (Include the correct units.) The presence of air resistance reduces the magnitude of the downward acceleration, and the effect of air resistance increases as the speed of the object increases. You repeat the experiment, but this time with a tennis ball as the object being dropped. Air resistance now has a noticeable effect on the data. (b) Is the final speed for a given release height higher than, lower than, or the same as when you ignored air resistance? (c) Is the graph of the release height versus the square of the final speed still a straight line? Sketch the qualitative shape of the graph when air resistance is present.

Two identical 15.0 - kg balls, each 25.0 cm in diameter, are suspended by two 35.0 - cm wires (Fig. P5.85). The entire apparatus is supported by a single 18.0 - cm wire, and the surfaces of the balls are perfectly smooth. (a) Find the tension in each of the three wires. (b) How hard does each ball push on the other one?

CP BIO Stress on the Shin Bone. The compressive strength of our bones is important in everyday life. Young’s modulus for bone is about $\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{10}}\mathbf{Pa}$. Bone can take only about a 1.0%change in its length before fracturing.

1. What is the maximum force that can be applied to a bone whose minimum cross-sectional area is $\mathbf{3}\mathbf{.}\mathbf{00}{\mathbf{cm}}^{\mathbf{2}}$? (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point.)
2. Estimate the maximum height from which aman could jump and not fracture his tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 s, and assume that the stress on his two legs is distributed equally.