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
The maximum speed of car with which it can go around a level road of radius 10 m is (coefficient of friction between the road and tyre is 0.5)
(g = 9.8 m/s²)
The maximum speed on a level road is limited by static friction: u=v²/rg
Given μs=0.5, g=9.8m/s², and r=10m.
Substituting these values: V²max=(0.5)(9.8)(10)= 49 = 7m/s.
Therefore, the maximum speed is 7 m/s.
A cyclist paddling at a speed of 10 m/s on a level road takes a sharp circular turn of radius 10 m without reducing the speed. The angle made by cyclist with vertical is
Using the formula
and given values
m/s,
m, and
m/s², we get:
Thus,
. The cyclist leans at 45° or π/4 with the vertical.
A particle moves in a circular path so that its distance travel varies with time \(t\) as \(s = 3t^2 + 6t\). Then its acceleration at \(t = 1\text{ sec.}\) is (radius of path is \(12\text{ m}\)) –
Speed is \(v = \frac{ds}{dt} = 6t + 6\). At \(t = 1\text{ s}\), \(v = 12\text{ m/s}\). Tangential acceleration is \(a_t = \frac{dv}{dt} = 6\text{ m/s}^2\). Centripetal acceleration is \(a_c = \frac{v^2}{R} = \frac{12^2}{12} = 12\text{ m/s}^2\). Total acceleration is \(a = \sqrt{a_t^2 + a_c^2} = \sqrt{6^2 + 12^2} = 6\sqrt{5}\text{ m/s}^2\).
A particle start revolving on a circular path with constant angular acceleration \(\frac{\pi}{2}\text{ rad/sec}^2\). Then find number of cycles it will complete in first 12 seconds:
Angular displacement is \(\theta = \frac{1}{2}\alpha t^2 = \frac{1}{2} \left(\frac{\pi}{2}\right) (12)^2 = 36\pi\text{ rad}\). Number of cycles \(N = \frac{\theta}{2\pi} = \frac{36\pi}{2\pi} = 18\).
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.