Question

A point mass oscillates along the $x-axis$ according to the law $x = x_0 \,cos (\omega t - \pi/4)$. If the acceleration of the particle is written as $a = A cos(\omega t + \delta)$ then

• A. $A = x_0 , \delta = -\pi/4$
• B. $A = x_0 \omega^2, \delta = -\pi/4$
• C. $A = x_0 \omega^2 , \delta = -\pi/4$
• D. $A = x_0 \omega^2, \delta = 3\pi/4$

Answer 1

Answer: (D)

$A = x_0 \omega^2, \delta = 3\pi/4$

Answer 2

The displacement x = x0 cos (ω t - π/4)

Now, as it is known that velocity (v) = derivative of displacement = dx/dt = -x0ω sin(ωt-π/4)

Now, Acceleration = derivative of velocity = dv/dt = -x0ω2 cos (ω t - π/4)

This equation can also be expressed as– a = x0ω2 cos (π + ωt - π/4) = x0ω2 cos (ωt - 3π/4) ----1)

From the equation, it is given that, a= A cos (ωt + ẟ) —--2)

By comparing equations 1) and 2)

A = x0ω2 and ẟ = 3π/4

Periodic fluctuations in some measure, generally over time, around a central value are called oscillations. It is a measure of some recurring variation that changes over time.

It may be determined using an equilibrium condition.

Simply put, oscillation is when a body travels back and forth around the same location at a regular period of time. This motion is also referred to as oscillatory motion.

The mean or equilibrium position is the axis around which the body travels.

Similar to how oscillatory motion is sometimes referred to as vibratory motion, mechanical oscillations are also known as vibrations.

The motion of a pendulum's bob, the piston of a car's engine, the motion of a mass coupled to a spring, etc. are all examples of oscillatory motion.

Every periodic motion is an oscillatory motion, although not all periodic motions are oscillatory.