The mass of rod PA and QB are negligible and they are free to rotate about hinge points P and Q respectively in the horizontal plane. The block of mass ‘m’ is attached to springs. Springs are light and their stiffness are shown in the figure. The block is slightly displaced along x-axis then released. Then the time period of oscillation of the block?

Question

Answer

Problem Statement:

The mass of rod PA and QB are negligible and they are free to rotate about hinge points P and Q respectively in the horizontal plane. The block of mass ‘m’ is attached to springs. Springs are light and their stiffness are shown in the figure. The block is slightly displaced along x-axis then released. Then the time period of oscillation of the block? Explain in details.

Calculation:

Let the displacement of the block be x1 and x2 from the equilibrium position.

Since the system is in equilibrium, the force acting on the block is equal to the force acting on the spring.

Let K1 and K2 be the stiffness of the springs on the left and right sides of the block respectively.

Therefore, K1x1 = K2x2. This equation represents the equilibrium condition of the system.

Let T be the time period of oscillation of the block.

Using the equation of motion for simple harmonic motion, we get:

T = 2π√(m/K1+K2)

Explanation:

The system shown in the figure is a combination of two springs connected in parallel. When the block is displaced from equilibrium position and released, it starts oscillating. The forces acting on the block are the forces due to the two springs on either side of it. The stiffness of the springs is given by K1 and K2. The displacement of the block from the equilibrium position is x1 and x2 on the left and right sides respectively. The equilibrium condition of the system is K1x1 = K2x2. Using the equation of motion for simple harmonic motion, we can find the time period of oscillation of the block. Therefore, T = 2π√(m/K1+K2).

Similar IIT JAM Doubts

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.?

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

Top Courses for IIT JAM

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams