Homework Help is Here – Start Your Trial Now!
Engineering
Mechanical Engineering
RWS-1 Consider a bowling ball which is tossed down a bowling alley. For this problem, we consider the bowling ball to be a uniform sphere of mass M and radius R, with a moment of inertia given by 1=(2/5)MR². The moment the ball hits the ground (t=0), it is moving horizontally with initial linear speed vo, but it is not rotating (mo 0). Due to kinetic friction between the ground and the ball, it begins to rotate as it slides. The coefficient of kinetic friction is ju. As the ball slides along the lane, its angular speed steadily increases. At some point (time te), the "no-slip" condition kicks in, so that o = v/R. After this, the ball moves with a constant linear and angular speed. Solve all parts of this problem symbolically. a) Use Newton's second law to find an expression for the linear acceleration of the ball along the x-direction while the ball is slipping, a. The free body diagram of the ball is shown below. Your final expression should only involve the variables g and u. b) Use the rotational version of Newton's second law to find an expression for the angular acceleration of the ball along the z-direction while the ball is slipping, az. Your final expression should only involve the variables R, g, and uk. c) You should have found that the linear acceleration a. from part a) and the angular acceleration az from part b) are constant. This means that our familiar kinematic equations below will apply while the ball slides down the lane. At time te, the moment the ball begins to roll without slipping, we have oz =-V/R (the negative sign is due to the fact that the ball rotates clockwise, in the -z direction). Use this to find an expression for the time te. Your final expression should only involve the

RWS-1 Consider a bowling ball which is tossed down a bowling alley. For this problem, we consider the bowling ball to be a uniform sphere of mass M and radius R, with a moment of inertia given by 1=(2/5)MR². The moment the ball hits the ground (t=0), it is moving horizontally with initial linear speed vo, but it is not rotating (mo 0). Due to kinetic friction between the ground and the ball, it begins to rotate as it slides. The coefficient of kinetic friction is ju. As the ball slides along the lane, its angular speed steadily increases. At some point (time te), the "no-slip" condition kicks in, so that o = v/R. After this, the ball moves with a constant linear and angular speed. Solve all parts of this problem symbolically. a) Use Newton's second law to find an expression for the linear acceleration of the ball along the x-direction while the ball is slipping, a. The free body diagram of the ball is shown below. Your final expression should only involve the variables g and u. b) Use the rotational version of Newton's second law to find an expression for the angular acceleration of the ball along the z-direction while the ball is slipping, az. Your final expression should only involve the variables R, g, and uk. c) You should have found that the linear acceleration a. from part a) and the angular acceleration az from part b) are constant. This means that our familiar kinematic equations below will apply while the ball slides down the lane. At time te, the moment the ball begins to roll without slipping, we have oz =-V/R (the negative sign is due to the fact that the ball rotates clockwise, in the -z direction). Use this to find an expression for the time te. Your final expression should only involve the

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
See similar textbooks
icon
Related questions
Question
RWS-1 Consider a bowling ball which is tossed down a bowling alley. For this problem, we will consider the bowling ball to be a uniform sphere of mass M and radius R, with a moment of inertia given by I= (2/5)MR2. The moment the ball hits the ground (t 0), it is moving horizontally with initial linear speed vo, but it is not rotating (oo = 0). Due to kinetic friction between the ground and the ball, it begins to rotate as it slides. The coefficient of kinetic friction is uk. As the ball slides along the lane, its angular speed steadily increases. At some point (time to), the "no-slip" condition kicks in, so that o = v/R. After this, the ball moves with a constant %3D %3D linear and angular speed. Solve all parts of this problem symbolically. a) Use Newton's second law to find an expression for the linear acceleration of the ball along the x-direction while the ball is slipping, ax. The free body diagram of the ball is shown below. Your final expression should only involve the variables g and uk. b) Use the rotational version of Newton's second law to find an expression for the angular acceleration of the ball along the z-direction while the ball is slipping, az. Your final expression should only involve the variables R, g, c) You should have found that the linear acceleration a from part a) and the angular acceleration oa, from part b) are constant. This means that our familiar kinematic equations below will apply while the ball slides down the lane. At time te, the moment the ball begins to roll without slipping, we have oz =-Vx/R (the negative sign is due to the fact that the ball rotates clockwise, in the -z direction). Use this to find an expression for the time te. Your final expression should only involve the and Hk. variables vo, g, and V = Vox +at and , = Wo, + a_t d) Come up with an expression for the linear speed of the ball when no slip condition kicks in. Simply plug the time te and the linear acceleration ax into the kinematic equation for vx above. Your final expression should just be a numerical factor multiplied by vo. e) The kinetic energy of the ball when it begins to roll without slipping is less than the initial kinetic energy of the ball (at t 0). This is because kinetic friction acts on the ball as it slides down the lane, which removes mechanical energy. Come up with an expression the change in kinetic energy, which only depends on M and vo. Remember, the ball is a rigid body with both rotational and translational kinetic energy. f) Finally, calculate the percentage of the initial kinetic energy that is lost (change in kinetic energy divided by initial kinetic energy times 100). This will be a just number, which does not depend on any of the the variables in the problem. CM Vo
Transcribed Image Text:RWS-1 Consider a bowling ball which is tossed down a bowling alley. For this problem, we will consider the bowling ball to be a uniform sphere of mass M and radius R, with a moment of inertia given by I= (2/5)MR2. The moment the ball hits the ground (t 0), it is moving horizontally with initial linear speed vo, but it is not rotating (oo = 0). Due to kinetic friction between the ground and the ball, it begins to rotate as it slides. The coefficient of kinetic friction is uk. As the ball slides along the lane, its angular speed steadily increases. At some point (time to), the "no-slip" condition kicks in, so that o = v/R. After this, the ball moves with a constant %3D %3D linear and angular speed. Solve all parts of this problem symbolically. a) Use Newton's second law to find an expression for the linear acceleration of the ball along the x-direction while the ball is slipping, ax. The free body diagram of the ball is shown below. Your final expression should only involve the variables g and uk. b) Use the rotational version of Newton's second law to find an expression for the angular acceleration of the ball along the z-direction while the ball is slipping, az. Your final expression should only involve the variables R, g, c) You should have found that the linear acceleration a from part a) and the angular acceleration oa, from part b) are constant. This means that our familiar kinematic equations below will apply while the ball slides down the lane. At time te, the moment the ball begins to roll without slipping, we have oz =-Vx/R (the negative sign is due to the fact that the ball rotates clockwise, in the -z direction). Use this to find an expression for the time te. Your final expression should only involve the and Hk. variables vo, g, and V = Vox +at and , = Wo, + a_t d) Come up with an expression for the linear speed of the ball when no slip condition kicks in. Simply plug the time te and the linear acceleration ax into the kinematic equation for vx above. Your final expression should just be a numerical factor multiplied by vo. e) The kinetic energy of the ball when it begins to roll without slipping is less than the initial kinetic energy of the ball (at t 0). This is because kinetic friction acts on the ball as it slides down the lane, which removes mechanical energy. Come up with an expression the change in kinetic energy, which only depends on M and vo. Remember, the ball is a rigid body with both rotational and translational kinetic energy. f) Finally, calculate the percentage of the initial kinetic energy that is lost (change in kinetic energy divided by initial kinetic energy times 100). This will be a just number, which does not depend on any of the the variables in the problem. CM Vo
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Forms of Energy
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Elements Of Electromagnetics
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY
SEE MORE TEXTBOOKS