CP A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar to what is shown in Fig. 10.10. The pulley is a uniform disk with mass 10.0 kg and radius 30.0 cm and turns on frictionless bearings. You measure that the stone travels 12.6 m in the first 3.00 s starting from rest. Find (b) the tension in the wire.

Verified step by step guidance

1. Step 1: Calculate the acceleration of the stone. Since the stone starts from rest and travels 12.6 m in 3.00 s, use the kinematic equation for uniformly accelerated motion: \( s = ut + \frac{1}{2}at^2 \), where \( s \) is the distance traveled, \( u \) is the initial velocity (0 m/s in this case), \( t \) is the time, and \( a \) is the acceleration.
2. Step 2: Determine the angular acceleration of the pulley. Since the linear acceleration of the stone is related to the angular acceleration of the pulley by the radius of the pulley (\( a = r\alpha \)), solve for \( \alpha \) using the radius of the pulley and the linear acceleration calculated in Step 1.
3. Step 3: Calculate the net torque acting on the pulley. The net torque \( \tau \) is given by \( \tau = I\alpha \), where \( I \) is the moment of inertia of the pulley and \( \alpha \) is the angular acceleration. For a uniform disk, the moment of inertia is \( I = \frac{1}{2}MR^2 \), where \( M \) is the mass of the pulley and \( R \) is its radius.
4. Step 4: Relate the tension in the wire to the torque. The torque is also given by \( \tau = TR \), where \( T \) is the tension in the wire and \( R \) is the radius of the pulley. Set the two expressions for torque equal to each other and solve for \( T \).
5. Step 5: Substitute the values of \( \alpha \), \( R \), and \( I \) into the equation derived in Step 4 to find the tension in the wire.

Key Concepts

Newton's Second Law

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is crucial for analyzing the forces acting on the stone and the pulley system, allowing us to relate the tension in the wire to the acceleration of the stone.

Rotational Dynamics

Rotational dynamics involves the study of the motion of objects that rotate about an axis. In this scenario, the pulley acts as a rotating object, and its moment of inertia plays a key role in determining how the tension in the wire affects its angular acceleration, which is linked to the linear acceleration of the stone.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In this problem, kinematic equations can be used to determine the acceleration of the stone based on its displacement and time, which is essential for calculating the tension in the wire.