## Tuesday, January 17, 2017

### Oscillations Problems with Solution One

We are solving series of problems based on the concept of oscillations. Oscillation is a kind of motion where the body oscillates about a fixed point called mean position and all oscillatory motions are periodic. It means oscillatory motion is repeated at regular intervals of time. We need to understand that all oscillatory motions are periodic but all periodic motions are not oscillatory. If oscillatory motion is also satisfying a condition like displacement is directly proportional to acceleration and acceleration is always directed to wards the mean position, we call that kind of oscillatory motion as simple harmonic motion. Simple pendulum is one example that executes simple harmonic motion when it is sightly disturbed from its mean position. We have derived equation for displacement, velocity and acceleration for a body in simple harmonic motion.

Problem

To a body in a simple harmonic motion, velocity is represented as shown in the equation below. We need to measure maximum acceleration that the body can get in the given conditions.

Solution

We have all ready derived equation for the velocity of the particle in simple harmonic motion. We need to get the given equation in the terms of the standard equation and the problem can be solved as shown in the diagram below.

Problem

A particle starts from mean position to a new position and it is as shown in the diagram below. Its amplitude and time period is given to us in the problem. We need to find the displacement where the velocity is half of the maximum velocity.

Solution

We know that the particle in simple harmonic motion has maximum velocity at the mean position. As per the given problem at a given instant, velocity of the particle is half of that maximum. Taking that into consideration and substituting the data in the standard format, we can solve the problem as shown in the diagram below.

Problem

Two different particles are in simple harmonic motion and their displacements are represented  as the given equations of the problem. We need to find the resultant amplitude of the combination. Problem is as shown below.

Solution

When we add to oscillatory motion, we need to get a oscillatory motion. The resultant amplitude can be found using the vector addition equation and the solution is as shown in the diagram below.

Problem

A simple harmonic oscillator starts from extreme position and covers a half the displacement in a given time. We need to measure the further time it is going to take to reach the mean position and the problem is as shown in the diagram below.

Solution

As the particle is here starting from the mean position, we need to know that it has some initial phase that is ninety degree. We know that the particle takes one forth of the time period to reach from extreme to mean position and to measure the remaining time to cover half amplitude to, we need to subtract from it as shown in the diagram below.

Problem

Number of springs are connected in series as shown in the problem to a a given mass and the system is allowed to oscillate. We need to measure the time period of oscillation of that system.

Solution

We know that when the springs are in series, the force acting on all of them is same and the extension in the spring is different and it depends on the nature of the spring. Using the common formula for the time period of the system and further simplify the problem as shown in the diagram below.