A light rod AB of length ‘ l ’ which is smoothly pivoted at the end A as shown.What will be angular frequency (rad/sec) for small oscillation of block of mass ‘m’ in this arrangement shown. Neglect the mass of the rod?

Question

Answer

Calculation of Angular Frequency for Small Oscillation of Block in Arrangement

Derive the expression for angular frequency of small oscillations

To derive the expression for angular frequency of small oscillations, we need to consider the forces acting on the block. The block experiences two forces, gravitational force and tension force. The gravitational force acts vertically downwards and the tension force acts along the rod towards the pivot point A. We can resolve these forces along the radial and tangential directions.

Resolving the forces along the radial direction, we get:

T cos θ = mg

Resolving the forces along the tangential direction, we get:

T sin θ = ml ω^2

Where,

T = tension in the rod

θ = angle made by the rod with the vertical

m = mass of the block

g = acceleration due to gravity

l = length of the rod

ω = angular frequency

Dividing the two equations, we get:

tan θ = (l ω^2)/g

For small oscillations, we can assume that θ is very small. Therefore, we can approximate tan θ as θ. Hence, we get:

θ = (l ω^2)/g

Substituting this value of θ in the first equation, we get:

T = mg/cos θ = mg/(1+(l ω^2)/g^2)

Substituting this value of T in the second equation, we get:

ml ω^2 = mg tan θ = (mg l ω^2)/g^2

Cancelling mg from both sides, we get:

ω^2 = g/l

Therefore, the expression for angular frequency of small oscillations is:

ω = √(g/l)

Explain the significance of the result obtained

The result obtained is significant because it shows that the angular frequency of small oscillations depends only on the length of the rod and the acceleration due to gravity. It does not depend on the mass of the block or the tension in the rod. This is a fundamental result in oscillatory motion and is applicable to a wide range of physical systems.

Conclusion

Therefore, the angular frequency of small oscillations of a block attached to a light rod pivoted at one end is given by √(g/l).

Similar IIT JAM Doubts

Consider the arrangement as shown in figure. Spring constants are k1 and k2. All surfaces are frictionless. If block is slightly displaced, what is the angular frequency of oscillation?

A uniform rod of mass m and length l is free to rotate about a fixed axis through its end and perpendicular to its length . Find the period of small oscillations of the rod.?

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