A person starts from rest at the top of a large frictionless spherical surface,

Question

A person starts from rest at the top of a large frictionless spherical surface, and slides into the water below. At what angle (theta) does the person leave the surface? (Hint: When the person leaves the surface, the normal force is zero.) The diagram shows an unknown radius and theta. A person starts from rest at the top of a large frictionless spherical surface, and slides into the water below. At what angle (theta) does the person leave the surface? (Hint: When the person leaves the surface, the normal force is zero.) The diagram shows an unknown radius and theta.

Explanation / Answer

Let theta be the angle of the radius (to the person) measured from the horizontal. m g sin theta - N = m v^2 / R where N is the normal force When N = 0 the person slides off of the sphere. m g sin theta = m v^2 / R 1/2 m v^2 = m g h where h is the vertical distance the person descends h = R (1 - cos phi) where phi = 90 - theta or h = R (1 - sin theta) 1/2 m v^2 = m g R sin theta / 2 using the first equations m g h = m g R (1 - sin theta) = m g R sin theta / 2 1 - sin theta = sin theta / 2 sin theta = 2/3 theta = 41.8 deg

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