Conservation of Energy

AP Physics 1· difficulty 4/5

A small block slides down a frictionless ramp from height hh and enters a vertical circular loop of radius RR. What is the minimum starting height hh such that the block makes it around the loop without losing contact at the top?

  • A

    hmin=2Rh_{\min} = 2\,R

  • B

    hmin=1.5Rh_{\min} = 1.5\,R

  • C

    hmin=5Rh_{\min} = 5\,R

  • D

    hmin=2.5Rh_{\min} = 2.5\,R

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Explanation

At the top, minimum speed satisfies vtop2=gRv_{\text{top}}^2 = gR. Energy conservation: mgh=mg(2R)+12mvtop2=2mgR+12mgRmgh = mg(2R) + \tfrac{1}{2}m v_{\text{top}}^2 = 2mgR + \tfrac{1}{2}mgR. Hence h=2.5Rh = 2.5R.

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