Classwork Series and Exercises {Physics}: Energy of Simple Harmonic Motion

Introduction

Kinetic energy is the energy one possesses due to motion. When a pendulum bob swings, it experiences a movement of displacement. The change in position as a result of little application of force, causing motion called kinetic energy.

Kinetic energy (K.E) = ½ MV²

                                    = ½ MW² [r² – y²]

Given mass M, velocity V, distance y, and displacement r

Potential Energy: Is the energy possessed due to level or position. The potential or ability possessed makes it to be where it is, i.e. the position is the distance above the ground. At the end of oscillation the bob of the pendulum has maximum PE and zero KE as the bob, moves back towards the centre of oscillation. The PE becomes smaller while the KE becomes higher.

Change in PE equals the gain in KE

Kinetic energy + Potential energy = Total SHM energy

KE = ½ MV² ½ MW² r² – ½ MW² [r² – y²] = ½ mw²y²

Rules about PE and KE

Rule 1: When KE is zero, PE is maximum

Rule 2: When KE is maximum, PE is zero

Rule 3: Maximum KE = ½ mw²r² = maximum PE.

A mass oscillating on a spring in a gravity-free vacuum has two sorts of energy: kinetic energy and elastic (potential) energy. Kinetic energy is given by:

E_k = ½ mv²

Elastic energy, or elastic potential energy, is given by:

E_e = ½ kx²

So, the total energy stored by the oscillator is:

SE = ½ (mv² + kx²)

This total energy is constant. However, the proportions of this energy which are kinetic and elastic change over time, since v and x change with time. If we give a spring a displacement, it has no kinetic energy, and a certain amount of elastic energy. If we let it go, that elastic energy is all converted into kinetic energy, and so, when the mass reaches its initial position, it has no elastic energy, and all the elastic energy it did have has been converted into kinetic energy. As the mass continues to travel, it is slowed by the spring, and so the kinetic energy is converted back into elastic energy – the same amount of elastic energy as it started out. The nature of the energy oscillates back and forth, but the total energy is constant.

If the mass is oscillating vertically in a gravitational field, the situation gets more complicated since the spring now has gravitational potential energy, elastic potential energy and kinetic energy. However, it turns out (if you do the maths) that the total energy is still constant, although the equilibrium position will be lower.

Types of Oscillation

Free Oscillation: This is an oscillation in which the amplitude of vibration is constant, e.g. a body hanging freely from one end of a spiral spring, displaced through a small distance and released. It oscillates to and fro about a mean position.

Damp Oscillation: In damp oscillation, the amplitude of vibration of the body diminishes exponentially, e.g. a body attached to one of a spiral spring and vibrating in a liquid.

Forced Vibration and Resonance

Forced Vibration: Is said to occur when a system is acted upon by a periodic force, whose frequency is not equal to its natural frequency. A louder sound is heard when a drum is beaten harder by the drummer. The force applied by the drummer in contact with the drum makes the drum to vibrate at the same frequency. The vibration of the force is called forced vibration

Resonance:  This is an effect caused by a vibrating body, causing another body to vibrate from its natural frequency. It is an applied forced vibration which occurs when the forcing frequency is equal to the natural frequency.

Examples of resonance are: (i) resonance in a tube (ii) resonance in a ripple tank (iii) resonance in boxes with an open end (iv) electrical radio receiver (v) dark line in a continuous spectrum.

QUESTIONS

Lets see how much you’ve learnt, attach the following answers to the comment below

1. A mass M hangs in equilibrium on a spring. M is made to oscillate about the equilibrium position by pulling it down 10 cm and releasing it. The time for M to travel back to the equilibrium position for the first time is 0.50 s. Which line, A to D, is correct for these oscillations?

                                                Amplitude/cm                                                  Period/s

A                                             10                                                                           1.0

B                                             10                                                                           2.0

C                                             20                                                                           2.0

D                                             20                                                                           1.0

2. A particle oscillates with undamped simple harmonic motion. Which one of the following statements about the acceleration of the oscillating particle is true? It is least when the speed is greatest B. It is always in the opposite direction to its velocity C. It is proportional to the frequency D. It decreases as the potential energy increases
3. Which one of the following statements is not true for a body vibrating in simple harmonic motion when damping is present? The damping force is always in the opposite direction to the velocity B. The damping force is always in the opposite direction to the acceleration C. The presence of damping gradually reduces the maximum potential energy of the system D. The presence of damping gradually reduces the maximum kinetic energy of the system
4. The frequency of a body moving with simple harmonic motion is doubled. If the amplitude remains the same, which one of the following is also doubled?   the time period   B. the total energy  C.  the maximum velocity  D. the maximum acceleration
5. A body moves with simple harmonic motion of amplitude A and frequency b/2 . What is the magnitude of the acceleration when the body is at maximum displacement?   A. zero  B.  4p²Ab²   C. Ab²  D. 4p²Ab^-2

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