Concept explainers
A rigid, massless rod has three particles with equal masses attached to it as shown in Figure P11.37. The rod is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through the point P and is released from rest in the horizontal position at t = 0. Assuming m and d are known, find (a) the moment of inertia of the system of three particles about the pivot, (b) the torque acting on the system at t = 0, (c) the
Figure P11.37
(a)
The moment of inertia of the system of three particles about the pivot.
Answer to Problem 11.49AP
The moment of inertia of the system of three particles about the pivot is
Explanation of Solution
The mass of three particles is
The formula to calculate moment of inertia is,
The distance of the particle 1 from point P is,
The distance of the particle 2 from point P is,
Substitute
Conclusion:
Therefore, the moment of inertia of the system of three particles about the pivot is
(b)
The torque acting on the system at
Answer to Problem 11.49AP
The torque acting on the system at
Explanation of Solution
Consider that the whole weight,
The formula to calculate torque is,
Substitute
Conclusion:
Therefore, the torque acting on the system at
(c)
The angular acceleration of the system at
Answer to Problem 11.49AP
The angular acceleration of the system at
Explanation of Solution
The formula to calculate angular acceleration is,
Substitute
Conclusion:
Therefore, the angular acceleration of the system at
(d)
The linear acceleration of the particle 3 at
Answer to Problem 11.49AP
The linear acceleration of the particle 3 at
Explanation of Solution
The formula to calculate linear acceleration is,
Substitute
Conclusion:
Therefore, the linear acceleration of the particle 3 at
(e)
The maximum kinetic energy of the system.
Answer to Problem 11.49AP
The maximum kinetic energy of the system is
Explanation of Solution
Because the axle is fixed, no external work is performed on the system of the earth and three particles, so the total mechanical energy is conserved.
The rotation kinetic energy is maximum when rod has swung to a vertical orientation with the centre of gravity directly under the axle.
The expression for the energy is,
Conclusion:
Therefore, the maximum kinetic energy of the system is
(f)
The maximum angular speed reached by the rod.
Answer to Problem 11.49AP
The maximum angular speed reached by the rod is
Explanation of Solution
In the vertical orientation, the rod has the greatest rotational kinetic energy.
The expression for the kinetic energy is,
Substitute
Conclusion:
Therefore, the maximum angular speed reached by the rod is
(g)
The maximum angular momentum of the system.
Answer to Problem 11.49AP
The maximum angular momentum of the system is
Explanation of Solution
The expression for the angular momentum is,
Substitute
Conclusion:
Therefore, the maximum angular momentum of the system is
(h)
The maximum speed of particle 2.
Answer to Problem 11.49AP
The maximum speed of particle 2 is
Explanation of Solution
The expression for the speed is,
Substitute
Conclusion:
Therefore, the maximum speed of particle 2 is
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Chapter 11 Solutions
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