Advertisements
Advertisements
Question
What is meant by simple harmonic oscillation? Give examples and explain why every simple harmonic motion is a periodic motion whereas the converse need not be true.
Advertisements
Solution
Simple harmonic motion is a special type of oscillatory motion in which the acceleration or force on the particle is directly proportional to its displacement from a fixed point and is always directed towards that fixed point. In one dimensional case, let x be the displacement of the particle and ax be the acceleration of the particle, then
ax ∝ x .................(1)
ax = −bx ...............(2)
where b is a constant which measures acceleration per unit displacement and dimensionally it is equal to T-2.
By multiplying by the mass of the particle on both sides of equation (1) and from Newton’s second law, the force is
Fx = −kx ….........(3)
where k is a force constant which is defined as force per unit length. The negative sign indicates that displacement and force (or acceleration) are in opposite directions. This means that when the displacement of the particle is taken towards the right of equilibrium position (x takes positive value), the force (or acceleration) will point towards equilibrium (towards left) and similarly, when the displacement of the particle is taken towards left of equilibrium position (x takes negative value), the force (or acceleration) will point towards equilibrium (towards right). This type of force is known as restoring force because it always directs the particle executing simple harmonic motion to restore to its original (equilibrium or mean) position. This force (restoring force) is central and attractive whose center of attraction is the equilibrium position.
In order to represent in two or three dimensions, we can write using vector notation
`vec"F" = -"k"vec("r")` ...(4)
where `vec"r"` is the displacement of the particle from the chosen origin. Note that the force and displacement have a linear relationship. This means that the exponent of force `vec"F"` and the exponent of displacement `vec"r"` are unity. The sketch between cause (magnitude of force `vec|"F"|`) and effect (magnitude of displacement `vec|"r"|`) is a straight line passing through the second and fourth quadrant.
By measuring slope `1/"k"`, one can find the numerical value of force constant k.
APPEARS IN
RELATED QUESTIONS
Can simple harmonic motion take place in a non-inertial frame? If yes, should the ratio of the force applied with the displacement be constant?
The motion of a particle is given by x = A sin ωt + B cos ωt. The motion of the particle is
A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.
A simple pendulum of length l is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with and acceleration a0(b) is going down with an acceleration a0 and (c) is moving with a uniform velocity.
A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.
A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of 45° with the X-axis. The two motions are given by x = x0 sin ωt and s = s0 sin ωt. Find the amplitude of the resultant motion.
Define the time period of simple harmonic motion.
What is an epoch?
A body oscillates with SHM according to the equation x = 5 cos `(2π"t" + π/4)`. Its instantaneous displacement at t = 1 sec is:
A body having specific charge 8 µC/g is resting on a frictionless plane at a distance 10 cm from the wall (as shown in the figure). It starts moving towards the wall when a uniform electric field of 100 V/m is applied horizontally toward the wall. If the collision of the body with the wall is perfectly elastic, then the time period of the motion will be ______ s.
