Question

A hollow sphere with outer radius R, inner radius R/2 and mass M rolls down a slope that is inclined at an angle ? to the horizontal. The sphere starts from rest and rolls without slipping a distance L down the slope. Choose x to be measured down the slope, starting from a point O where the sphere starts from rest. Measure ? anticlockwise from the normal to the slope as shown in the diagram, so that x = R? as the sphere rolls down the slope. Choose unit vectors i pointing along Ox down the slope, and j at an angle ?/2 anticlockwise (the same sense as increasing ?), as shown in the diagram. (a) Draw a force diagram showing all of the forces acting on the sphere, and mark the point of action of each force. [3] (b) Apply Newton's second law to the sphere to obtain an equation for the linear acceleration ? of the sphere in terms of the magnitude F of the friction force at the point of contact, and other parameters. [2] (c) Calculate the torque of each force about the axis of rotation of the sphere. [3] (d) Apply the equation of relative rotational motion to the sphere to show that the acceleration of the sphere is 70/101 g sin ?, where g is the magnitude of the acceleration due to gravity. [6] (e) Calculate the time t for the sphere to roll a distance L down the slope without slipping, in terms of known constants. [3]

          A hollow sphere with outer radius R, inner radius R/2 and mass M rolls down a slope that is inclined at an angle ? to the horizontal. The sphere starts from rest and rolls without slipping a distance L down the slope. Choose x to be measured down the slope, starting from a point O where the sphere starts from rest. Measure ? anticlockwise from the normal to the slope as shown in the diagram, so that x = R? as the sphere rolls down the slope. Choose unit vectors i pointing along Ox down the slope, and j at an angle ?/2 anticlockwise (the same sense as increasing ?), as shown in the diagram.

(a) Draw a force diagram showing all of the forces acting on the sphere, and mark the point of action of each force. [3]
(b) Apply Newton's second law to the sphere to obtain an equation for the linear acceleration ? of the sphere in terms of the magnitude F of the friction force at the point of contact, and other parameters. [2]
(c) Calculate the torque of each force about the axis of rotation of the sphere. [3]
(d) Apply the equation of relative rotational motion to the sphere to show that the acceleration of the sphere is 70/101 g sin ?, where g is the magnitude of the acceleration due to gravity. [6]
(e) Calculate the time t for the sphere to roll a distance L down the slope without slipping, in terms of known constants. [3]

A hollow sphere with outer radius R, inner radius R/2 and mass M rolls down a slope that is inclined at an angle ? to the horizontal. The sphere starts from rest and rolls without slipping a distance L down the slope. Choose x to be measured down the slope, starting from a point O where the sphere starts from rest. Measure ? anticlockwise from the normal to the slope as shown in the diagram, so that x = R? as the sphere rolls down the slope. Choose unit vectors i pointing along Ox down the slope, and j at an angle ?/2 anticlockwise (the same sense as increasing ?), as shown in the diagram.

(a) Draw a force diagram showing all of the forces acting on the sphere, and mark the point of action of each force. [3]
(b) Apply Newton's second law to the sphere to obtain an equation for the linear acceleration ? of the sphere in terms of the magnitude F of the friction force at the point of contact, and other parameters. [2]
(c) Calculate the torque of each force about the axis of rotation of the sphere. [3]
(d) Apply the equation of relative rotational motion to the sphere to show that the acceleration of the sphere is 70/101 g sin ?, where g is the magnitude of the acceleration due to gravity. [6]
(e) Calculate the time t for the sphere to roll a distance L down the slope without slipping, in terms of known constants. [3]

Added by Edward P.

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