A hanger of mass m is attached to a light string that is wrapped around the rim

Question

A hanger of mass m is attached to a light string that is wrapped around the rim of a wheel of radius r. The wheel is released, allowing the weight to fall and cause the wheel to accelerate rotationally. In the experiment, the rotational acceleration ? is measured. The acceleration of gravity is g (defined as a positive quantity), and the moment of inertia is I. Work out a formula for I by following the stteps given below. All answers are formulas, and should contain only symbols from the following set: hanger mass m (type "m"), acceleration of gravity g (type "g"), moment of inertia I (type "I"), radius r (type "r"), linear acceleration of the falling weight a (type "a"), string tension T (type "T"), and rotational acceleration ? (type "alpha"). Use the previewer ("eye" icon) to check the formatting of your answer before you submit it. Check this link for help on entering formulas: Click here for help with symbolic formatting. Important: Regard m, g, T, r and I as positive quantities. The accelerations a and ? may be positive or negative, depending on direction. In the coordinate system assumed "positive" means up for linear quantities and counterclockwise for angular quantities. Write both sides of Newton's 2nd law for the falling weight: the net force, given in terms of the tension T and the gravity force mg should equal the weight's mass times it acceleration. Sum of forces mass

Explanation / Answer

FOLLOW THIS in these equations t and a are the two variables t= ma + mg put t in 2nd eq. (mg +ma) r = -Ia/r mg + ma = -Ia/r2 a= -mgr2/(mr2+ I ) check if I=0 ; a= -g if I->infinity ; a =0 a=-ra=-mgr2/(mr2+ I ) => I + mr2 = mgr2 / ra I = mr2 (g/ra - 1)

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