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A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R, as shown below. R Image Description Follow steps below to find the angle at which the body will leave the sphere. a. First, we need an expression for speed v as a function of angle 0. Using conservation of energy, find an expression for v in terms of given quantities, m, R, 0, and/or gravitational acceleration g. Hint for (a) Start by setting up conservation of energy equation. Use the fact that the block starts from rest at the top, and use the geometric relationships to express change in height in terms of 0. When the block has slid down to angle 0, you have v = b. Next piece we need relates to the forces on the block. Before the block loses contact, at angle 0, what is the magnitude of the radial component of the net force? Answer in terms of given quantities above, and use N for the magnitude of normal force. Hint for (b) The radial component of the net force is Fnet,r = c. We now put (a) and (b) together. The net force you found in (b) is centripetal force and is also equal to Fnet.r = mv² /R. Set up this equation, plug in v from (a) above, and for the angle 0 at which the block loses contact, let N = 0 (for "losing contact"). Solve for cos(0) in the resulting expression and answer below. Hint for (c). cos(0)

A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R, as shown below. R Image Description Follow steps below to find the angle at which the body will leave the sphere. a. First, we need an expression for speed v as a function of angle 0. Using conservation of energy, find an expression for v in terms of given quantities, m, R, 0, and/or gravitational acceleration g. Hint for (a) Start by setting up conservation of energy equation. Use the fact that the block starts from rest at the top, and use the geometric relationships to express change in height in terms of 0. When the block has slid down to angle 0, you have v = b. Next piece we need relates to the forces on the block. Before the block loses contact, at angle 0, what is the magnitude of the radial component of the net force? Answer in terms of given quantities above, and use N for the magnitude of normal force. Hint for (b) The radial component of the net force is Fnet,r = c. We now put (a) and (b) together. The net force you found in (b) is centripetal force and is also equal to Fnet.r = mv² /R. Set up this equation, plug in v from (a) above, and for the angle 0 at which the block loses contact, let N = 0 (for "losing contact"). Solve for cos(0) in the resulting expression and answer below. Hint for (c). cos(0)

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A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R, as shown below. R Image Description Follow steps below to find the angle at which the body will leave the sphere. a. First, we need an expression for speed v as a function of angle 0. Using conservation of energy, find an expression for v in terms of given quantities, m, R, 0, and/or gravitational acceleration g. Hint for (a) Start by setting up conservation of energy equation. Use the fact that the block starts from rest at the top, and use the geometric relationships to express change in height in terms of 0. When the block has slid down to angle 0, you have v = b. Next piece we need relates to the forces on the block. Before the block loses contact, at angle 0, what is the magnitude of the radial component of the net force? Answer in terms of given quantities above, and use N for the magnitude of normal force. Hint for (b) The radial component of the net force is Fnet,r = c. We now put (a) and (b) together. The net force you found in (b) is centripetal force and is also equal to Fnet.r = mv² / R. Set up this equation, plug in v from (a) above, and for the angle 0 at which the block loses contact, let N = 0 (for "losing contact"). Solve for cos(0) in the resulting expression and answer below. Hint for (c). cos(0)
Transcribed Image Text:A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R, as shown below. R Image Description Follow steps below to find the angle at which the body will leave the sphere. a. First, we need an expression for speed v as a function of angle 0. Using conservation of energy, find an expression for v in terms of given quantities, m, R, 0, and/or gravitational acceleration g. Hint for (a) Start by setting up conservation of energy equation. Use the fact that the block starts from rest at the top, and use the geometric relationships to express change in height in terms of 0. When the block has slid down to angle 0, you have v = b. Next piece we need relates to the forces on the block. Before the block loses contact, at angle 0, what is the magnitude of the radial component of the net force? Answer in terms of given quantities above, and use N for the magnitude of normal force. Hint for (b) The radial component of the net force is Fnet,r = c. We now put (a) and (b) together. The net force you found in (b) is centripetal force and is also equal to Fnet.r = mv² / R. Set up this equation, plug in v from (a) above, and for the angle 0 at which the block loses contact, let N = 0 (for "losing contact"). Solve for cos(0) in the resulting expression and answer below. Hint for (c). cos(0)
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