Concept explainers
A tiny solid ball
Fig. 10-12
(a)
The speed of the ball as it passes through the point B as shown in Fig. 10-12.
Answer to Problem 55SP
Solution:
Explanation of Solution
Given data:
The moment of inertia of the solid ball is
The ball is released at point A. Therefore, the value of the initial velocity at that point is zero.
The movement of ball is as shown in Fig. 10-12;itfollows a circular path of radius
Formula used:
Write the expression for calculating thepotential energy of an object:
Here,
Write the expression for the total kinetic energy of a rolling object:
Here,
Write the expression for total energy of system:
Here,
Write the expression for conservation of energy:
Here,
Write the expression for linear velocity in terms of angular velocity:
Here,
Explanation:
Draw the arrangement of the system shown in Fig 1:
Consider the point B be the reference point, that is, the height of the ball is measured with respect to this point.
The expression forpotential energy of the ball at point A is
Here,
For point A, substitute
The expression forpotential energy of ball at point B is
Here,
For point B, substitute
The expression for the total kinetic energy of theball at point A is
Here,
Since the ball is released from rest at point A, its initial velocity is zero.
For point A, substitute
The expression for linear velocity in terms of angular velocity at point B is
Here,
At point B, substitute
Solve for
The expression for the total kinetic energy of ball at point B is
Here,
For point B, substitute
The expression for total energy of the system at point A is
Here,
At point A, substitute
The expression for the total energy of the system at point B is
Here,
At point B, substitute
The expression for conservation of energy for the point of observation A and point B is
Substitute
Solve for
Take square root both sides:
Conclusion:
The speed of the ball as it passes through the point B is
(b)
The speed of the ball as it passes through the point Cas shown in Fig. 10-12.
Answer to Problem 55SP
Solution:
Explanation of Solution
Given data:
The moment of inertia of the solid ball is
The ball is released at point A that is the velocity of the ball at this point is zero.
The movement of ball is shown in Fig. 10-12;it follows a circular path of radius
Formula used:
Write the expression for calculating the potential energy of an object:
Here,
Write the expression for total kinetic energy of a rolling object:
Here,
Write the expression for total energy of a system:
Here,
Write the expression for conservation of energy:
Here,
Write the expression for linear velocity in terms of angular velocity:
Here,
Explanation:
Let point B be the reference point, that is, the height of the ball is measured with respect to this point.
Recall the total energy at point A from subpart (a):
The expression forpotential energy of ball at point C is
Here,
The expression for height of point C is
Here,
Substitute
For point C, substitute
The expression for total kinetic energy of a rolling object at point C is
Here,
The expression for linear velocity in terms of angular velocity is
Here,
At point C, substitute
Solve for
For point C, substitute
The expression for total energy of the system at point C is
Here,
At point C, substitute
The expression for conservation of energy for the point of observation A and C is
Substitute
Solve for
Further, solve for
Conclusion:
The speed of the ball as it passes through the point C is
Want to see more full solutions like this?
Chapter 10 Solutions
Schaum's Outline of College Physics, Twelfth Edition (Schaum's Outlines)
- The precession angular velocity of a gyroscope is 1.0 rad/s. If the mass of the rotating disk is 0.4 kg and its radius is 30 cm, as well as the distance from the center of mass to the pivot, what is the rotation rate in rev/s of the disk?arrow_forwardIf the torque acting on a particle about an axis through a certain origin is zero, what can you say about its angular momentum about that axis?arrow_forwardCalculate the rotational kinetic energy of a 12-kg motorcycle wheel if its angular velocity is 120 rad/s and its inner radius is 0.280 m and outer radius 0.330 m.arrow_forward
- Two spheres, one hollow and one solid, are rotating with the same angular speed around an axis through their centers. Both spheres have the same mass and radius. Which sphere, if either, has the higher rotational kinetic energy? (a) The hollow I sphere, (b) The solid sphere, (c) They have the same kinetic energy.arrow_forwardA 40.0-kg solid cylinder is rolling across a horizontal surface at a speed of 6.0 m/s. How much work is required to stop it?arrow_forwardAn automobile engine can produce 200Nm of torque. Calculate the angular acceleration produced if 95.0 of this torque is applied to the drive shaft, axle, and rear wheels of a car, given the following information. The car is suspended so that the wheels can turn freely. Each wheel acts like a 15.0-kg disk that has a 0.180-m radius. The walls of each tire act like a 2.00-kg annular ring that has inside radius of 0.180 m and outside radius of 0.320 m. The tread of each tire acts like a 10.0-kg hoop of radius 0.330 m. The 14.0-kg axle acts like a rod that has a 2.00-cm radius. The 30.0-kg drive shaft acts like a rod that has a 3.20-cm radius.arrow_forward
- Repeat Example 10.15 in which the stick is free to have translational motion as well as rotational motion.arrow_forwardIf global warming continues, its likely that some ice from the polar ice caps of the Earth will melt and the water will be distributed closer to the equator. If this occurs, would the length of the day (one rotation) (a) increase, (b) decrease, or (c) remain the same?arrow_forwardWhich of the entries in Table 10.2 applies to finding the moment of inertia (a) of a long, straight sewer pipe rotating about its axis of symmetry? (b) Of an embroidery hoop rotating about an axis through its center and perpendicular to its plane? (c) Of a uniform door turning on its hinges? (d) Of a coin turning about an axis through its center and perpendicular to its faces?arrow_forward
- A cylinder of mass 10.0 kg rolls without slipping on a horizontal surface. At a certain instant, its center of mass has a speed of 10.0 m/s. Determine (a) the translational kinetic energy of its center of mass, (b) the rotational kinetic energy about its center of mass, and (c) its total energy.arrow_forwardA constant torque is applied to a rigid body whose moment of inertia is 4.0kgm2 around the axis of rotation. If the wheel starts from rest and attains an angular velocity of 20.0 rad/s in 10.0 s, what is the applied torque?arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningUniversity Physics Volume 1PhysicsISBN:9781938168277Author:William Moebs, Samuel J. Ling, Jeff SannyPublisher:OpenStax - Rice UniversityCollege PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
- College PhysicsPhysicsISBN:9781285737027Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningGlencoe Physics: Principles and Problems, Student...PhysicsISBN:9780078807213Author:Paul W. ZitzewitzPublisher:Glencoe/McGraw-HillPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning