Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 73 | Concept Questions The drawing shows a rectangular piece of wood. The forces applied to corners B and D have the same magnitude and are directed parallel to the long and short sides of the rectangle. An axis of rotation is shown perpendicular to the plane of the rectangle at its center. (a) Relative to this axis, which force produces the torque with the greater magnitude? (b) A force is to be applied to corner A, directed along the short side of the rectangle. The net torque produced by the three forces is zero. How is the force at corner A directed—from A toward B or away from B? Give your reasoning.
Problem The magnitudes of the forces at corners B and D are each 12 N. The long side of the rectangle is twice as long as the short side. Find the magnitude and direction of the force applied to corner A.

| 74 | Concept Questions The wheels, axle, and handles of a wheelbarrow weigh 60.0 N. The load chamber and its contents weigh 525 N. The drawing shows these two forces in two different wheelbarrow designs. To support the wheelbarrow in equilibrium, the man’s hands apply a force to the handles that is directed vertically upward. Consider a rotational axis at the point where the tire contacts the ground, directed perpendicular to the plane of the paper. (a) For which design is there a greater total torque from the 60.0- and 525-N forces? (b) For which design does the torque from the man’s force need to be greater? (c) For which design does the man’s force need to be greater? Account for your answers.

Problem Find the magnitude of the man’s force for both designs. Be sure that your answers are consistent with your answers to the Concept Questions.

| 75 | Concept Questions Part a of the drawing shows a uniform horizontal beam attached to a vertical wall by a frictionless hinge and supported from below at an angle by a brace that is attached to a pin. The forces that keep the beam in equilibrium are its weight and the forces applied to the beam by the hinge and the brace. (a) Part b of the drawing shows the weight and the force from the brace, while assuming that the hinge applies a force that is vertical. Can these three forces keep the beam in equilibrium? (b) Part c of the drawing is like part b, except that it assumes the hinge applies a force that is horizontal rather than vertical. Can the three forces in part c keep the beam in equilibrium? (c) Part d of the drawing assumes that the hinge applies a force that has both a vertical component and a horizontal component . Can the forces in part d keep the beam in equilibrium? Justify your answers.
Problem The brace makes an angle with respect to the beam, which has a weight of 340 N. Find the magnitudes of the forces , , and .

| 76 | Concept Questions Multiple-Concept Example 10 provides one model for solving this type of problem. Two wheels have the same mass and radius. One has the shape of a hoop and the other the shape of a solid disk. Each wheel starts from rest and has a constant angular acceleration with respect to a rotational axis that is perpendicular to the plane of the wheel at its center. Each makes the same number of revolutions in the same time. (a) Which wheel, if either, has the greater angular acceleration? (b) Which, if either, has the greater moment of inertia? (c) To which wheel, if either, is a greater net external torque applied? Explain your answers.
Problem The mass and radius of each wheel are 4.0 kg and 0.35 m, respectively. Each wheel starts from rest and turns through an angle of 13 rad in 8.0 s. Find the net external torque that acts on each wheel. Check to see that your answers are consistent with your answers to the Concept Questions.

| 77 | Concept Questions Two thin rods of length L are rotating with the same angular speed (in rad/s) about axes that pass perpendicularly through one end. Rod A is massless but has a particle of mass 0.66 kg attached to its free end. Rod B has a mass 0.66 kg, which is distributed uniformly along its length. (a) Which has the greater moment of inertia—rod A with its attached particle or rod B? (b) Which has the greater rotational kinetic energy? Account for your answers.
Problem The length of each rod is 0.75 m, and the angular speed is 4.2 rad/s. Find the kinetic energies of rod A with its attached particle and of rod B. Make sure your answers are consistent with your answers to the Concept Questions.

| 78 | Concept Questions As seen from above, a playground carousel is rotating counterclockwise about its center on frictionless bearings. A person standing still on the ground grabs onto one of the bars on the carousel very close to its outer edge and climbs aboard. Thus, this person begins with an angular speed of zero and ends up with a nonzero angular speed, which means that he underwent a counterclockwise angular acceleration. (a) What applies the force to the person to create the torque causing this acceleration? What is the direction of this force? (b) According to Newton’s action-reaction law, what can you say about the direction of the force applied to the carousel by the person and about the nature (clockwise or counterclockwise) of the torque that it creates? (c) Does the torque identified in part (b) increase or decrease the angular speed of the carousel?
Problem The carousel has a radius of 1.50 m, an initial angular speed of 3.14 rad/s, and a moment of inertia of 125 kg · m2. The mass of the person is 40.0 kg. Find the final angular speed of the carousel after the person climbs aboard. Verify that your answer is consistent with your answers to the Concept Questions.

| 79 | Concept Questions The drawing shows two identical systems of objects; each consists of three small balls (masses m1, m2, and m3 ) connected by massless rods. In both systems the axis is perpendicular to the page, but it is located at a different place, as shown. (a) Do the systems necessarily have the same moments of inertia? If not, why not? (b) The same force of magnitude F is applied to the same ball in each system (see the drawing). Is the magnitude of the torque created by the applied force greater for system A or for system B? Or is the magnitude the same in the two cases? Explain. (c) The two systems start from rest. Will system A or system B have the greater angular speed at the same later time? Or will they have the same angular speeds? Justify your answer.

Problem The masses of the balls are , and . The magnitude of the force is F = 424 N. (a) For each of the two systems, determine the moment of inertia about the given axis of rotation. (b) Calculate the torque (magnitude and direction) acting on each system. (c) Both systems start from rest, and the direction of the force moves with the system and always points along the 4.00-m rod. What is the angular velocity of each system

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.