Problem 1

A small block of mass

m

is released from rest at the top of a fixed, smooth spherical dome of radius

R

and slides down its outside; take gravitational acceleration

g

as constant. The angle

\theta

is measured from the vertical through the top to the radius through the block, with

0\le\theta<\pi/2

. Neglect the block's size and drag. (a) Use energy conservation to find its speed as a function of

\theta

. (b) Write Newton's second law radially and state the loss-of-contact condition. (c) Find the departure angle and explain why it is independent of

m,R,g

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

mg\cos\theta-N=\frac{mv^2}{R}

problems.proof.analysis

Choose inward as positive. The inward gravity component is $mg\cos\theta$ , while the dome's normal force points outward; their difference gives centripetal acceleration $v^2/R$ .