Concept explainers
A slender rod of length l and mass m is pivoted about a point C located at a distance b from its center G. It is released from rest in a horizontal position and swings freely. Determine (a) the angular velocity of the rod as it passes through a vertical position if b = l/2, (b) the distance b for which the angular velocity of the rod as it passes through a vertical position is maximum, (c) the corresponding values of its angular velocity and of the reaction at C using the value of b calculated.
Fig. P17.16
(a)
Find the angular velocity of the rod when
Answer to Problem 17.16P
The angular velocity of the rod when
Explanation of Solution
Show the free-body diagram of the given condition as in Figure 1.
Find the mass moment of inertia of the slender rod
Here, the mass of the slender rod is m and the length of the slender rod is l.
Position 1 (Horizontal position):
The angular velocity
The velocity
Find the total kinetic energy in the horizontal position
Substitute 0 for
Positon 2 (Vertical position):
Find the velocity of the slender rod
Find the total kinetic energy in the vertical position
Substitute
Find the work done
Here, the acceleration due to gravity is g.
Write the equation of work and energy for the system using the equation.
Substitute 0 for
Therefore, the angular velocity of the rod when
(b)
Find the distance b for which the angular velocity of rod as it passes through a vertical position is maximum.
Answer to Problem 17.16P
The distance b for which the angular velocity of the rod is maximum in vertical position is
Explanation of Solution
Position 1 (Horizontal position):
Show the free-body diagram of the horizontal position as in Figure 2.
Find the mass moment of inertia of the slender rod
The angular velocity
The velocity
Find the total kinetic energy in the horizontal position
Substitute 0 for
The elevation (h) of the pivot C is zero.
Find the total potential energy
Substitute 0 for h.
Positon 2 (Vertical position):
Show the free-body diagram of the vertical position as in Figure 3.
Find the velocity of the slender rod
Find the total kinetic energy in the vertical position
Substitute
The elevation of the pivot C is
Find the total potential energy
Substitute b for h.
Write the equation of conservation of energy using the equation.
Substitute 0 for
Integrate the angular velocity with respect to b and equate to zero.
Therefore, the distance b for which the angular velocity of the rod is maximum in vertical position is
(c)
Find the angular velocity where the vertical position is maximum and the reaction at pivot C.
Answer to Problem 17.16P
The angular velocity corresponding to the maximum vertical position is
The reaction at pivot C is
Explanation of Solution
Refer to the calculation of part (b):
Substitute
Therefore, the angular velocity corresponding to the maximum vertical position is
Show the free-body diagram of the slender rod as in Figure 4.
Find the normal acceleration
Substitute
The value of tangential acceleration is
Resolve the vertical component of forces.
Take moment about point C as follows;
Therefore,
Resolve the horizontal component of forces.
Find the resultant reaction at point C using the relation.
Substitute 0 for
Therefore, the reaction at pivot C is
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