Concept explainers
(a)
The couple
Answer to Problem 18.102P
The couple
Explanation of Solution
Given information:
Angular velocity of disk in z-direction is
The figure is represented below.
Figure (1)
Write the equation for the mass of the disk.
Here, weight of disk is
Write the expression for the angular momentum about point
Here, mass moment of inertia about the x-axis is
Write the expression for the angular velocity of disk in x direction.
Substitute
Here,
Write the expression for angular velocity in vector form.
Write the expression for rate of angular velocity of the reference frame
Write the expression for rate of total angular velocity.
Substitute
Write the expression for Matrix multiplication of the vector product for Equation (8).
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 the matrix multiplication of the vector product for Equation (13).
Write the expression for the acceleration of the mass centre of the disk.
Write the expression for the matrix multiplication of the vector product for Equation (15).
Write the expression for the the sum of the forces acting on the system.
Write the expression for the force in terms of mass and acceleration.
Substitute
Here, force at
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 between
Write the expression for
Here, distance from the centre of disk to point
Substitute
Write the expression for the matrix multiplication for vector product for equation (24).
Write the expression for the moment about
Here, length of
Write the expression for the matrix multiplication for the vector product for equation (26).
Here
The sum of the moment at
Substitute
Compare the coefficients of the unit vector of
Compare the coefficients of the unit vector of
Compare the coefficients of the unit vector of
Substitute
Substitute
Calculation:
Substitute
Substitute values of
Thus value of couple
Conclusion:
The value of couple is
(b)
The dynamic reaction at
The dynamic reaction at
Answer to Problem 18.102P
The dynamic reaction at
The dynamic reaction at
Explanation of Solution
Given information:
Write the expression for the angular velocity in terms of time in y-direction.
Here time is
Calculation:
Substitute
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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