Concept explainers
A cam of mass M is in the shape of a circular disk of diameter 2R with an off-center circular hole of diameter R is mounted on a uniform cylindrical shaft whose diameter matches that of the hole (Fig. P1 3.78). a. What is the rotational inertia of the cam and shaft around the axis of the shaft? b. What is the rotational kinetic energy of the cam and shaft if the system rotates with angular speed ω around this axis?
(a)
The rotational inertia of the cam and shaft around the axis of the shaft.
Answer to Problem 78PQ
The rotational inertia of the cam and shaft around the axis of the shaft is
Explanation of Solution
Write the equation for the rotational inertia of the cam and shaft around the axis of the shaft.
Here,
Rotational inertia of the cam is the difference of the rotational inertia of the solid disk about an axis
Write the equation for the rotational inertia of the cam.
Here,
The rotational inertia of the solid disk bout an axis
Write the equation for
Here,
Write the equation for the rotational inertia of the disk about its center of mass.
Put the above equation in equation (III).
With half the radius, the cut away small disk has one-quarter the face area, one-quarter the volume and one-quarter the mass of the original solid disk.
Write the expression for the ratio of the mass of the small disk to the mass of the original solid disk.
Here,
Rewrite the above equation for
Write the equation for the rotational inertia of the small disk about an axis through its center of mass.
Here,
Put equation (V) in the above equation.
Put equations (IV) and (VI) in equation (II).
Write the equation for the mass of the cam.
Here,
Put equation (V) in the above equation.
Multiply and divide the right hand side of equation (VII) with
Put equation (VIII) in the denominator of the above equation.
Write the equation for the rotational inertia of the shaft.
Here,
Conclusion:
Put equations (IX) and (X) in equation in equation (I).
Therefore, the rotational inertia of the cam and shaft around the axis of the shaft is
(b)
The rotational kinetic energy of the cam and shaft if the system rotates with angular speed
Answer to Problem 78PQ
The rotational kinetic energy of the cam and shaft if the system rotates with angular speed
Explanation of Solution
Write the equation for the rotational kinetic energy of the cam and the shaft.
Here,
Write the equation for
Put equation (IX) in the above equation.
Write the equation for
Put equation (X) in the above equation.
Conclusion:
Put equations (XII) and (XIII) in equation (XI).
Therefore, the rotational kinetic energy of the cam and shaft if the system rotates with angular speed
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Chapter 13 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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