sliding snowball from hemispherical surface

Question

A snowball starts sliding down at the top of a hemispherical surface. The mass of the snowball is changing as a function of angle θ with the vertical as follows m = m0e −α cos(θ) where m0 and α are positive constants. Find the work done by the gravitational force when the mass reaches point P.

Answer 1

I am assuming that this question came from chegg.com, and I will use Figure 5

on that website.

Let W(θ) be the work performed by gravity by the time the snowball reaches n arbitrary angle θ.

Let dW(θ) be its increment due to the angle increasing by dθ.

As usual, dθ is small enough that to first order we can assume the snowball

to loose no mass while traversing the angle dθ.

dW(θ) = m(θ) g R dθ sin(θ)

where

m(θ) g is the gravitation force at angle θ

R dθ is distance traveled by the snowball through angle dθ

sin(θ) represents the downward component of the snowball's trajectory

Plugging in the function m(θ) and rearranging

dW(θ) = g R m0 sin(θ) e^(-α cos(θ)) dθ

W is obtained by integrating the above.

The indefinite integral is

W(θ) = (g R m0 / α) e^(-α cos(θ))

The work performed by the time the snowball reaches the point P is the definite

integral between 0 and Θ, which is

(g R m0 / α) (e^(-α cos(Θ)) - e^(-α cos(0))) = (g R m0 / α) (e^(-α cos(Θ)) - e^-a)