Question

A pendulum of mass m and length ℓ is suspended from the ceiling of a trolley which has a constant acceleration a in the horizontal direction as shown in the figure. Work done by the tension is ______.

(In the frame of the trolley)

 

Options

• `"mg"/(tan theta)`[cos [tan^-1(a/g)] - 1] 

• `"mg"/(tan theta)`[sin [tan^-1 (a/g)] - 1] 

• `"mg"/(cos theta)`[sin [tan^-1 (a/g)] - 1] 

• `"mg"/(cos theta)`[cos [tan^-1 (a/g)] - 1] 

Answer 1

A pendulum of mass m and length ℓ is suspended from the ceiling of a trolley which has a constant acceleration a in the horizontal direction as shown in the figure. Work done by the tension is `underlinebb("mg"/(cos theta)[cos [tan^-1 (a"/"g)] - 1])`.

Explanation:

From the equation of motion,

T sinθ = ma 

T cosθ = mg

∴ tanθ = `"a"/"g"`

dW = -T sinθ.ldθ

W = `int_0^(tan^-1("a""/""g"))` -T sinθ.ldθ

= Tl `[cos theta]_0^(tan^-1("a""/""g"))`

= Tl [cos [tan^-1(a/g)] - 1]

= `"mg"/(cos theta)`[cos [tan^-1(a/g)] - 1]