Question

What minimum horizontal speed should be given to the bob of a simple pendulum of length l so that it describes a complete circle?

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Suppose the bob is given a horizontal speed v₀ at the bottom and it describes a complete vertical circle. Let its speed at the highest point be v. Taking the gravitational potential energy to be zero at the bottom, the conservation of energy gives.

$\frac{1}{2}m{v}_0^2=\frac{1}{2}m{v}^2+2mgl$

or, $m{v}^2=m{v}_0^2-4mgl$ --------------(i)

The forces acting on the bob at the highest point are mg due to the gravity and T due to the tension in the string. The resultant force towards the centre is, therefore, mg + T. As the bob is moving in a circle, its acceleration towards the centre is $\frac{v^2}{l}$. Applying Newton's second law and using (i),
mg + T = $m\frac{v^2}{l}=\frac{1}{l}(m{v}_0^2-4mgl)$

Or $m{v}_0^2=5mgl+Tl$.

Now, for $v_0$ to be minimum, T should be minimum. As the minimum value of T can be zero, for minimum speed,

$m{v}_0^2=5mgl$ or, $v_0=\sqrt{5gl}$.