A wheel having diameter of 3 m starts from rest and accelerates uniformly to an angular velocity of 210 rpm in 5 seconds. Angular acceleration of the wheel is :
The angular velocity of earth about its axis of rotation is :
Angular speed ω = 2π / T = 2π / (60×60×24) rad /sec
Two bodies of masses 10 kg and 5 kg moving on concentric orbits of radii R and r such that their period of revolution are same. The ratio of their centripetal acceleration is :
As time period of revolution is same for both the particles angular speed will be equal for both.
Centripetal Acceleration is given by ω²r.
so a 1 / a 2 = R/r
A stone of mass m is tied to a string of length l and rotated in a circle with a constant speed v. If the string is released, the stone flies :
In Circular motion velocity of the particle is always directed along tangent. So when string is released object moves tangentially.
A stone of mass 1 kg tied to one end of a string 1.0 m long is revolved in a horizontal circle at the rate of 10/π revolution per second. Calculate the tension of the string ?
Tension force in the string will provide required centripetal force
T = mω²r= 1× (10/π ×2π)²×1= 400 N
A particle moves in a circle of radius 5 m with constant speed and time period 2π s. The acceleration of the particle is :
As speed of the particle is constant only centripetal acceleration will act on the object.
Centripetal Acceleration = ω²R
a c = (2π/2π)× 5= 5 m/s²
Assertion: In uniform circular motion tangential acceleration of particle is zero.
Reason: In uniform circular motion net force on particle is always directed towards centre of circular path.
Tangential acceleration is zero because speed is constant. The net force is centripetal, which is directed towards the center. Thus, both are true but the Reason is not the explanation of the Assertion.
A particle moves in a circle of radius \(10\text{ cm}\) with constant speed and time period \(\pi\text{ s}\). The acceleration of the particle is :
The angular frequency is \(\omega = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2\text{ rad/s}\). The centripetal acceleration is \(a = \omega^2 R = 2^2 \times 10 = 40\text{ cm/s}^2\).
Assertion (A): If a body is in state of uniform circular motion then its velocity and acceleration both are varying.
Reason (R): If magnitude of velocity is \(v\) and radius of uniform circular motion is \(r\) then magnitude of acceleration is \(v^2/r\).
In uniform circular motion, velocity (vector) and acceleration (vector) are varying due to changing direction. So (A) is true. The magnitude of centripetal acceleration is \(a_c = v^2/r\). So (R) is true. However, (R) describes the magnitude, not the reason for vector variation. Thus, (R) is not the correct explanation of (A).
Assertion (A): In uniform circular motion, angular acceleration is zero.
Reason (R): In uniform circular motion, acceleration is constant.
In uniform circular motion, angular speed is constant, making angular acceleration \(alpha = 0\). So (A) is true. However, linear acceleration (centripetal acceleration) continuously changes direction, so it is not constant. Thus (R) is false.