Question

A rigid body is rotating with an angular velocity ω = a - bt, where a and b are constants and t is the time. Then the total angle subtended by the body before coming to rest is:
1. \(\frac{(a-b)a}{2}\)
2. \(\frac{a^2}{2b}\)
3. \(\frac{a^2 -b^2}{2b}\)
4. \(\frac{a^2 -b^2}{2a}\)

Answer 1

Correct Answer - Option 2 : \(\frac{a^2}{2b}\)

The correct answer is option 2) i.e. \(\frac{a^2}{2b}\)

CONCEPT:

• Angular velocity is the rate of change of angular displacement with time for motion on a curved path.
  • While moving on a curved path, displacement is the change in angle about the point of rotation or curved motion.

Angular velocity, \(ω =\frac{angle}{time} =\frac{dθ}{dt}\)

CALCULATION:

Given that:

Angular velocity, ω = a - bt

When the body comes to rest, ω = 0

⇒ a - bt = 0

⇒ t = a/b      

Therefore, the body comes to rest at t = a/b

We know, \(ω =\frac{dθ}{dt}\)

\(⇒ a-bt =\frac{dθ}{dt}\)

\(⇒ (a-bt)dt=dθ\)

Total angle subtended, θ = ∫dθ 

\(⇒∫dθ = \int_{0}^{a/b} (a-bt)dt\)

\(⇒∫dθ = [at - \frac{bt^2}{2}]_0 ^{a/b}\)

\(⇒∫dθ = a(\frac{a}{b}) - \frac{b(\frac{a}{b})^2}{2}\)

\(⇒∫dθ = \frac{a^2}{2b}\)

\(∴ θ =\frac{a^2}{2b}\)