A wheel of radius R, mass M, and moment of inertia I is mounted on a frictionless, horizontal axle as in the figure. A light cord wrapped around the wheel supports an object of mass m. When the wheel is released, the object accelerates downward, the cord unwraps off the wheel, and the wheel rotates with an angular acceleration. Find expressions for the angular acceleration of the wheel, the translational acceleration of the object, and the tension in the cord. M SOLVE IT Conceptualize Imagine that the object is a bucket in an old- fashioned water well. It is tied to a cord that passes around a cylinder equipped with a crank for raising the bucket. After the bucket has been raised, the system is released and the bucket accelerates downward while the cord unvwinds off the cylinder. Categorize We apply two analysis models here. The object is modeled as a particle under a net force. The wheel is modeled as a rigid object under a net torque. Analyze The magnitude of the torque acting on the wheel about its axis of rotation is r = TR, where Tis the force exerted by the cord on the rim of the wheel. (The gravitational force exerted by the Earth on the wheel and the normal force exerted by the axle on the wheel both pass through the axis of rotation and therefore An abject hangs from a cord wrapped around a wheel. produce no torque.) From the rigid object under a net torque model, write the equation: (1) --- Solve for a and substitute the net torque: (1) a = From the particle under a net force model, apply Newton's second law to the motion of the object, taking the downward direction to be positive: E; = mg - T= ma Solve for the acceleration a: (2) mg - T Equations (1) and (2) have three unknowns: a, a, and T. Because the object and wheel are connected by a cord that does not slip, the translational acceleration of the suspended object is equal to the tangential acceleration of a point on the wheel's rim. Therefore, the angular acceleration a of the wheel and the translational acceleration of the object are related by a = Ra. Use this fact together with Equations (1) and (2): (3) a- Ra = TR mg m Solve for the tension T. (Use the following as necessary: m, g. R, and I.): (4) T= Substitute Equation (4) into Equation (2) and solve for a. (Use the following as necessary: m, g, R, and 1.): (5) a= Use a = Ra and Equation (5) to solve for a. (Use the following as necessary: m, g. R, and I.): MASTER IT HINT: GETTING STARTED Suppose the wheel (R = 20.0 cm, M = 2.80 kg) is rotated at a constant rate so that the mass m = 0.785 kg has an upward speed of 3.65 m/s when it reaches a point P. At that moment, the wheel is released to rotate on its own. It starts slowing down and eventually reverses its direction due to the downward tension of the cord. What is the maximum height, h, the mass will rise above the point P?