Concept explainers
A 2.5-kg homogeneous disk of radius 80 mm rotates at the constant rate
(a)
The couple applied to shaft to produce acceleration.
Answer to Problem 18.104P
The couple applied to shaft to produce acceleration is
Explanation of Solution
Given information:
Mass of homogeneous disk is
The figure is represented below.
Figure (1)
Write the expression for the angular velocity of disk in x direction.
Write the expression for the total angular velocity of the disk
Here
Substitute,
Write the expression for the angular momentum about point
Here, mass moment of inertia about the x-axis is
Substitute
Write the expression for angular velocity in vector form of shaft
Write the expression for rate of angular velocity of the reference frame
Here,
Write the expression for rate of total angular velocity.
Substitute
Write the expression for Matrix multiplication of the vector product for Equation (10).
Write the expression for the mass moment of inertia about the y-direction.
Here mass of the disk is
Write the expression for the mass moment of inertia about the z- direction.
Substitute
Write the expression for the velocity of mass centre of the disk.
Here, velocity of mass centre is
Write the expression for
Here, horizontal distance is
Substitute
Write the expression for the matrix multiplication of the vector product for Equation (17).
Write the expression for the acceleration of the mass centre of the disk.
Substitute
Write the expression for the matrix multiplication of the vector product for Equation (20).
Write the expression for the the sum of the forces acting on the system.
Here, force at
Write the expression for the force in terms of mass and acceleration.
Substitute
Substitute
Compare the coefficients of the unit vector of
Compare the coefficients of the unit vector of
Write the expression for the rate of angular momentum about
Here, distance of
Write the expression for
Here distance from point
Substitute
Write the expression for the matrix multiplication for vector product for equation (30).
Write the expression for the moment about
Here, moment couple when system is at rest is
Write the expression for the matrix multiplication for the vector product for equation (32).
The sum of the moment at
Substitute
Compare the coefficients of the unit vector of
Calculation:
Substitute values of
Thus value of couple
Conclusion:
The couple applied to shaft to produce acceleration is
(b)
The dynamic reaction at
The dynamic reaction at
Answer to Problem 18.104P
The dynamic reactions at
The dynamic reactions at
Explanation of Solution
Given information:
Compare the coefficients of the unit vector of
Compare the coefficients of the unit vector of
Substitute
Substitute
Write the expression for the angular velocity in terms of time in y-direction.
Write the expression for the angular velocity in terms of time in y-direction
Calculation:
Substitute values of
Substitute values of
Hence, dynamic reaction at
Substitute values of
Substitute values of
Hence, dynamic reaction at
Conclusion:
The dynamic reactions at
The dynamic reactions at
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Chapter 18 Solutions
Vector Mechanics For Engineers
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