Question

establish a relation between linear acceleration and angular acceleration

Answer 1

Consider a body revolving in a circular path of radius ‘r’ with angular velocity ‘ω’.

The relation between linear and angular velocity is,

v = rω

=> dv/dt = d(rω)/dt = rdω/dt + ωdr/dt

=> dv/dt = rdω/dt  [since, r is a constant]

=> a = r α

This is the relation between linear acceleration and angular acceleration.

hope this will help u...

Answer 2

The relation between linear acceleration and angular acceleration is: a= αω

• Linear acceleration: When the velocity of an item going straight forward changes over the course of a certain amount of time, it is said to be accelerating. Depending on whether the velocity is accelerating or decelerating, acceleration can either be positive or negative. The linear acceleration may be better understood by considering how a vehicle moves. The velocity is determined by the speedometer in the car.
• A terminal bus can start abruptly when pushed, as one may have noticed. The reason for this is because when the lift first starts, it pushes upward. The acceleration is brought on by the change in velocity in this instance. As a result, the rate of change of an object's velocity will be used to characterise acceleration.

• Angular acceleration: The time rate of change of angular velocity is referred to as angular acceleration in physics. Due to the fact that there are two different types of angular velocity—spin angular velocity and orbital angular velocity—there are also two different types of angular acceleration, referred to as spin angular acceleration and orbital angular acceleration, respectively. The terms "orbital angular acceleration" and "spin angular acceleration" describe the angular acceleration of a point particle about a fixed origin and, respectively, the angular acceleration of a rigid body about its centre of rotation.

• Relation between linear acceleration and angular acceleration:

$a= \frac{dv}{dt}$

$\alpha = \frac{dw}{dt}$

$\frac{a}{\alpha } = \frac{dv}{dt} *\frac{dt}{dw} \\or, \frac{a}{\alpha } = \frac{dv}{dw} \\or, \frac{a}{\alpha } = \frac{d(rw)}{dw} \\or, \frac{a}{\alpha } = r. \frac{dw}{dw}\\or, \frac{a}{\alpha } = r \\or, a = \alpha r$

Hence, the relation between linear acceleration and angular acceleration is: a= αω

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