Home
Class 11
PHYSICS
A wheel rotates with an angular accelera...

A wheel rotates with an angular acceleration given by `alpha =4at^3 - 3 bt^3`, where `t` is the time and `a` and `b` ae constants. If the wheel has initial angular speed `omega_0`, write the equations for the (a) angular speed (b) angular displacement.

Text Solution

Verified by Experts

(a) Since `alpha=(d omega)/(dt)rArrd omega=alphadt`
Integrating both sides, we get
`int_(omega_(0))^(omega)d omega=int_(0)^(t)alphadt=int_(0)^(t)(4alphat^(3)-3bt^(2))dt`
`omega-omega_(0)=4a(t^(4))/4-3b(t^(3))/3rArromega=omega_(0)+at^(4)-br^(3)`
(b) Since `omega=(dtheta)/(dt) rArrd theta=omegadt`
On integrating both sides, we get
`int_(0)^(theta)d theta=int_(0)^(t)omegadt=int_(0)^(t)(omega_(0)+at^(4)-bt^(3))dt`
`rArrtheta=omega_(0)t+(at^(5)/5-bt^(4)/4)`
Promotional Banner

Similar Questions

Explore conceptually related problems

The angular displacement of a particle is given by theta = t^3 + 2t +1 , where t is time in seconds. Its angular acceleration at t=2s is

A dust particle lying at the rim of a wheel has an angular speed of 22 rad/s. Find the time period.

Angular acceleration alpha of a body is given by the relation a = 4at^3 -3bt^2 . If initial angular velocity of the body is omega_0 , then its velocity at time t will be

Starting from rest a wheel rotates with uniform angular acceleration 2 pi rad s^(-2) . After 4s ,if the angular acceleration ceases to act,its angular displacement in the next 4s is?

A child's top is spun with angular acceleration alpha=5t^(3)-4t , with t in seconds and alpha in radians per second-squared. At t = 0, the top has angular velocity 5 rad/s, and a reference line on it is at angular position theta=2 rad. (a) Obtain an expression for the angular velocity omega(t) of the top. That is, find an expression that explicitly indicates how the angular velocity depends on time. (b) Obtain an expression for the angular position theta(t) of the top.

A wheel is rotating with uniform angular velocity vecomega . What is the nature of the graph between tangential speed at different points of the wheel and their radial distances from the axis?

The balance wheel of a watch vibrates with an angualr amplitude pi radians and a period of 0.5s. Find a) the maximum angular speed of the wheel, b) the angular speed of the wheel when its displacement is pi/2 radians and c) the angular acceleration of the wheel when its displacement is pi/4 radians. [hint: use formulae of linear SHM for the angular SHM of the wheel]

Starting from rest a wheel rotates with uniform angular acceleration 2 pi rad s^(-2) .After 4s ,if the angular acceleration ceases to act,its angular displacement in the next 4s is?

A solid body rotates about a stationary axis so that its angular velocity depends on the rotation angle varphi as omega=omega_0-avarphi , where omega_0 and a are positive constants. At the moment t=0 the angle varphi=0 . Find the time dependence of (a) the rotation angle, (b) the angular velocity.