Physic Labs

Problem 1

A small block of mass mm is released from rest at the top of a fixed, smooth spherical dome of radius RR and slides down its outside; take gravitational acceleration gg as constant. The angle θ\theta is measured from the vertical through the top to the radius through the block, with 0≤θ<π/20\le\theta<\pi/2. Neglect the block's size and drag. (a) Use energy conservation to find its speed as a function of θ\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,gm,R,g.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
12mv2=mgR(1−cos⁡θ)\frac12mv^2=mgR(1-\cos\theta)
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The smooth normal force is perpendicular to velocity and does no work, so lost gravitational potential becomes kinetic energy. The block starts from rest, with no initial kinetic energy.

problems.proof.pitfall. The vertical drop is R(1−cos⁡θ)R(1-\cos\theta), not Rsin⁡θR\sin\theta.