Question

A small sphere is hung by a string from the ceiling of a van. When the van is stationary, the sphere hangs vertically. However, when the van accelerates, the sphere swings backward so that the string makes an angle of
θ
with respect to the vertical.
(a) Derive an expression for the magnitude a of the acceleration of the van in terms of the angle
θ
and the magnitude g of the acceleration due to gravity.
(b) Find the acceleration of the van when
θ=10.0∘
(c) What is the angle
θ
when the van moves with a constant velocity?

Answer 1

Answer with Explanation:

We are given that

String makes an angle w.r.t  vertical=$\theta=$

a.We have to derive an expression for the magnitude of the acceleration of the van in terms of the angle $\theta$ and magnitude g of the acceleration due to gravity.

According to newton's second law

$T sin\theta=ma$

$Tcos\theta=mg$

$\frac{Tsin\theta}{Tcos\theta}=\frac{ma}{mg}$

$tan\theta=\frac{a}{g}$

$a=gtan\theta$

b.$\theta=10^{\circ}$

$g=9.8 m/s^2$

$a=9.8\times tan10^{\circ}=1.73 m/s^2$

c.Velocity=Constant

We have to find the angle $\theta$

$a=0$

$0=9.8tan\theta$

$tan\theta=0$

$tan\theta=tan0$

$\theta=0^{\circ}$