Problem 96 - University Physics with Modern Physics

Question

A capstan is a rotating drum or cylinder over which a rope or cond slides in order to provide a great amplification of the rope's tension while keeping both ends free (Fig. 17.33). Since the added tension in the rope is due to friction, the capstan generates thermal energy. (a) If the difference in tension between the two ends of the rope is 520.0 \(\mathrm{N}\) and the capstan has a diameter of 10.0 \(\mathrm{cm}\) and turns once in 0.900 \(\mathrm{s}\) , find the rate at which thermal energy is generated. Why does the number of turns not matter? (b) If the capstan is made of iron and has mass 6.00 \(\mathrm{kg}\) , at what rate does its temperature rise? Assume that the temperature in the capstan is uniform and that all the thermal energy generated flows into it.

Step-by-Step Solution

Answer

(a) 181.5 W; number of turns doesn't matter. (b) Temperature rise rate: 0.0676 °C/s.

Step 1: Determine the Angular Velocity

First, calculate the angular velocity of the capstan. The radius \( r \) is half of the diameter, so \( r = 0.05 \) m. Since the capstan turns once in 0.900 s, the angular velocity \( \omega \) is given by: \[ \omega = \frac{2\pi}{0.900} \approx 6.98 \text{ rad/s} \]

Step 2: Calculate Torque Due to the Tension Difference

The torque \( \tau \) exerted by the tension difference on the capstan is \( \tau = (T_2 - T_1) \times r \), where \( T_2 - T_1 = 520.0 \) N and \( r = 0.05 \) m. Therefore, the torque is: \[ \tau = 520.0 \times 0.05 = 26.0 \text{ Nm} \]

Step 3: Calculate the Power Generating Thermal Energy

The power \( P \) generating thermal energy is the product of torque and angular velocity: \[ P = \tau \times \omega = 26.0 \times 6.98 \approx 181.5 \text{ W} \] This is the rate at which thermal energy is generated. The number of turns does not matter because power involves torque and angular velocity, both of which remain constant regardless of the number of completed turns.

Step 4: Calculate the Heat Capacity of the Capstan

Use the specific heat capacity \( c \) of iron, which is approximately 448 J/kg·°C, and the mass \( m = 6.00 \) kg to calculate the heat capacity \( C \): \[ C = m \times c = 6.00 \times 448 = 2688 \text{ J/°C} \]

Step 5: Determine the Rate of Temperature Increase

The rate of temperature increase \( \frac{dT}{dt} \) of the capstan is given by the power divided by the heat capacity. Thus, \[ \frac{dT}{dt} = \frac{P}{C} = \frac{181.5}{2688} \approx 0.0676 \text{ °C/s} \]

Key Concepts

Angular Velocity

In the world of rotational motion, angular velocity plays a crucial role. It represents how fast something is rotating around a fixed point or axis. For a capstan, the angular velocity is an essential concept as it tells us how quickly the drum rotates. To find angular velocity, we use the formula \( \omega = \frac{\theta}{t} \), where \( \theta \) is the angle in radians and \( t \) is the time in seconds. However, for one full rotation, \( \theta \) becomes \( 2\pi \) radians.
In the exercise, given that the capstan makes one full turn in 0.900 seconds, we calculate the angular velocity as \( \omega = \frac{2\pi}{0.900} \). This results in approximately 6.98 rad/s.
This value indicates how quickly the capstan spins and is critical for further calculations like torque and thermal energy.

Torque

Torque is the measure of rotational force applied to an object, causing it to rotate. It’s the rotational equivalent of linear force and depends on two factors: the amount of force applied and the distance from the pivot point at which it is applied. For a capstan, the torque exerted is directly linked to the difference in rope tension on either side of the drum.
The formula to calculate torque is \( \tau = (T_2 - T_1) \times r \), where \( T_2 - T_1 \) is the tension difference and \( r \) is the radius of the capstan. Using the exercise values, we find that the torque is \( 26.0 \text{ Nm} \).
A consistent torque, like in this scenario, implies consistent rotational motion, which is crucial for calculating the power that generates thermal energy.

Thermal Energy Generation

When a capstan operates, the friction due to the rope rubbing against the drum converts mechanical energy into thermal energy. This process is measured as power, the rate at which energy is transformed.
The formula for power in terms of rotational motion is \( P = \tau \times \omega \), where \( \tau \) is torque and \( \omega \) is angular velocity. In our exercise, substituting in the values yields \( P = 26.0 \times 6.98 \approx 181.5 \text{ W} \).
This means that 181.5 watts of power are being generated as thermal energy. It's a steady rate, unaffected by the number of turns, because both torque and angular velocity are constant over the cycle.

Temperature Rise

The rise in temperature is a direct consequence of thermal energy generation. The rate at which the capstan’s temperature increases depends on the power converted into heat and the object’s heat capacity.
Mathematically, we express this as \( \frac{dT}{dt} = \frac{P}{C} \), where \( P \) is the power and \( C \) is the heat capacity. From our solution, \( \frac{dT}{dt} = \frac{181.5}{2688} \approx 0.0676 \text{ °C/s} \).
This indicates that the capstan’s temperature rises by approximately 0.0676 degrees Celsius per second, assuming uniform distribution of heat and no loss to the environment.

Specific Heat Capacity

Specific heat capacity is a fundamental property that tells us how much energy is required to raise the temperature of a unit mass of a substance by one degree Celsius. For the capstan, which is made of iron, it helps us understand how quickly its temperature rises.
The specific heat capacity for iron is around 448 J/kg·°C. To determine the capstan's ability to store thermal energy, we calculate the heat capacity, \( C = m \times c \), where \( m \) is mass and \( c \) is specific heat capacity. For our 6 kg iron capstan, \( C = 6.00 \times 448 = 2688 \text{ J/°C} \).
With this, we gauge how energy input relates to temperature rise, crucial for understanding the thermal dynamics during and after capstan operation.