Problem 12 - University Physics with Modern Physics

Question

You are unwinding a large spool of cable. As you pull on the cable with a constant tension, what happens to the angular acceleration and angular velocity of the spool, assuming that the radius at which you are extracting the cable remains constant and there is no friction force? a) Both increase as the spool unwinds. b) Both decrease as the spool unwinds. c) Angular acceleration increases, and angular velocity decreases. d) Angular acceleration decreases, and angular velocity increases. e) It is impossible to tell.

Step-by-Step Solution

Answer

Answer: Both increase as the spool unwinds.

Step 1: Identify the given information and the required variables

Given: - Constant tension on the cable - Radius of the spool remains constant - No friction force Required to find: The effect on angular acceleration and angular velocity as the spool unwinds.

Step 2: Calculate the torque acting on the spool

Since the tension T on the cable is constant, the torque τ acting on the spool can be calculated as: τ = T * R where R is the constant radius of the spool.

Step 3: Calculate the moment of inertia of the spool

The moment of inertia I of the spool can be calculated as: I = k * M * R^2 where M is the mass of thespool, and k is a constant depending on the shape of the spool. As the spool unwinds, its mass decreases. Thus, the moment of inertia I of the spool also decreases.

Step 4: Calculate the angular acceleration of the spool

Using Newton's second law for rotational motion, we have: τ = I * α where α is the angular acceleration of the spool. Since τ remains constant, as the moment of inertia I decreases, the angular acceleration α of the spool must increase to maintain the equality. Therefore, the angular acceleration of the spool increases as it unwinds.

Step 5: Calculate the angular velocity of the spool

The relationship between angular acceleration α, angular velocity ω, and time t is given by: ω = ω0 + α*t where ω₀ is the initial angular velocity. As we have established that angular acceleration α increases with time and that α is positive (since we are unwinding the spool), the final angular velocity ω will also increase with time.

Step 6: Determine the correct answer from the given options

Based on our analysis, both the angular acceleration and the angular velocity of the spool increase as it unwinds. Therefore, the correct choice is: a) Both increase as the spool unwinds.