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
v(θ)=2gR(1−cos⁡θ),cos⁡θleave=23,θleave=arccos⁡(23)≈48.2∘,vleave=2gR3\boxed{v(\theta)=\sqrt{2gR(1-\cos\theta)},\quad\cos\theta_{\rm leave}=\frac23,\quad\theta_{\rm leave}=\arccos\left(\frac23\right)\approx48.2^\circ,\quad v_{\rm leave}=\sqrt{\frac{2gR}{3}}}
problems.proof.analysis

The first expression gives speed at any angle while contact persists; the zero-normal condition gives departure angle arccos⁡(2/3)\arccos(2/3). Substituting it gives vleave=2gR/3v_{\rm leave}=\sqrt{2gR/3}; mass cancels from the angle condition.