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# HW14 - Prof I Biaggio Solutions HW 14 Physics 90 Fall 2006...

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Prof. I. Biaggio Solutions HW 14 Physics 90, Fall 2006 1-7 Solutions Homework 14 73 •• In this problem the mass m will accelerate downwards while pulling on the sphere. The pull on the sphere, through the string, will cause it to rotate faster and faster. The problem can be solved by relating the rate at which the angular velocity of the speed increases (via its moment of inertia) to the torque acting on it (which comes from the pull from the string), and realizing that the acceleration of the hanging object must be the same as the tangential acceleration of a point at the surface of the sphere where the string is wrapped. Picture the Problem The force diagram shows the forces acting on the sphere and the hanging object. The tension in the string is responsible for the angular acceleration of the sphere and the difference between the weight of the object and the tension is the net force acting on the hanging object. We can use Newton’s 2 nd law to obtain two equations in a and T that we can solve simultaneously. ( a )Apply Newton’s 2 nd law to the sphere and the hanging object: ! = = " # sphere 0 I TR (1) and ! = " = ma T mg F x (2) Substitute for I sphere and α in equation (1) to obtain: ( ) R a MR TR 2 5 2 = (3) Eliminate T between equations (2) and (3) and solve for a to obtain: m M g a 5 2 1 + = ( b ) Substitute for a in equation (2) and solve for T to obtain: M m mMg T 2 5 2 + =

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Prof. I. Biaggio Solutions HW 14 Physics 90, Fall 2006 2-7 74 •• This is an extension of the problem above. Again one can look at the rotational acceleration of the pulley (caused by the tensions in the stings), and again at the accelerations of the two hanging objects. All accelerations are related to each other because the objects are connected with a string. In this problem, note how the tensions in the left and right strings need to be different if the pulley has mass, but would tend to be equal for when its moment of inertia becomes negligible. Picture the Problem The diagram shows the forces acting on both objects and the pulley. By applying Newton’s 2 nd law of motion, we can obtain a system of three equations in the unknowns T 1 , T 2 , and a that we can solve simultaneously.
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HW14 - Prof I Biaggio Solutions HW 14 Physics 90 Fall 2006...

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