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`

`(Tsin\theta)/(Tcos\theta)=(ma)/(mg)`

`tan\theta=(a)/(g)`

`a=gtan\theta`

b.
`\theta=10^(\circ)`

`g=9.8 m/s^2`

`a=9.8* 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)`