Problem

The uniform slender rod has a mass m. If it is released from rest when θ = 0°, determine t...

The uniform slender rod has a mass m. If it is released from rest when θ = 0°, determine the magnitude of the reactive force exerted on it by pin B whenθ= 90°.

Step-by-Step Solution

Solution 1

Draw the schematic diagram of the rod as shown below:

Picture 3

Here, the centroid of the rod is.

Draw the free body diagram of the rod at any general position as shown below:

Picture 5

Calculate the distance from the centroid of the rod to point as shown below.

…… (1)

Here, the total length of the rod is and the distance from the centroid of the rod to point is.

Calculate moment of inertia of the rod about point B using the relation as shown below,

…… (2)

Here, the mass of the rod is , the moment of inertia of the rod about B is and the moment of inertia of the rod about the centroid is .

Substitute equation (1) in equation (2).

…… (3)

Consider the equilibrium of moments about the point B.

…… (4)

Here, the angular acceleration is and is the inclination of rod from the horizontal in clockwise direction.

Substitute equation (1) and equation (3) in equation (4).

…… (5)

Calculate the angular velocity of the rod at any general position. Consider the following kinematic equation.

…… (6)

Here, is the angular velocity of the rod.

Substitute for and integrate it.

…… (7)

Consider the free body diagram of the rod. Write the equilibrium of forces along

…… (8)

Here, the force acting at point along is .

Substitute equation (7) in equation (8).

…… (9)

Substitute in the above equation.

Consider the equilibrium of forces along

…… (10)

Here, the force acting at point along is.

Substitute equation (5) in equation (10).

…… (11)

Substitute in the above equation.

Calculate the magnitude of reactive force acting on the rod by using the equation,

Substitute and.

Hence, the magnitude of reactive force acting on the rod is.

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