Problem 54 - University Physics with Modern Physics

Question

A 20.0-kg rock is sliding on a rough, horizontal surface at 8.00 m/s and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is 0.200. What average power is produced by friction as the rock stops?

Step-by-Step Solution

Answer

156.86 W

Step 1: Calculate the Force of Friction

The force of friction \( f_k \) can be calculated using the formula \( f_k = \mu_k \cdot m \cdot g \), where \( \mu_k = 0.200 \) is the coefficient of friction, \( m = 20.0 \text{ kg} \) is the mass, and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity. So, \( f_k = 0.200 \times 20.0 \times 9.81 = 39.24 \text{ N} \).

Step 2: Calculate the Work Done by Friction

The work done by friction \( W \) is given by \( W = f_k \cdot d \), where \( d \) is the distance the rock travels until it stops. Since \( W \) also equals the change in kinetic energy, \( W = -\frac{1}{2} m v^2 \), we have \( W = -\frac{1}{2} \times 20.0 \times 8.00^2 = -640 \text{ J} \).

Step 3: Determine the Stopping Distance

The distance \( d \) the rock travels can be found by equating the expression for work: \( 39.24 \times d = 640 \). Solving for \( d \) gives \( d = \frac{640}{39.24} \approx 16.31 \text{ m} \).

Step 4: Calculate the Time Taken to Stop

First, use the acceleration \( a \) due to friction, given by \( a = \frac{f_k}{m} = \frac{39.24}{20.0} = 1.962 \text{ m/s}^2 \). Then, use the formula \( v_f = v_i + at \), with final velocity \( v_f = 0 \) and initial velocity \( v_i = 8.0 \text{ m/s} \), to solve for \( t \): \( 0 = 8.0 - 1.962 \cdot t \). This gives \( t \approx 4.08 \text{ s} \).

Step 5: Calculate Average Power Produced by Friction

Power \( P \) is the work done per unit time, so \( P = \frac{W}{t} = \frac{640}{4.08} \approx 156.86 \text{ W} \).

Key Concepts

Kinetic Friction

When two surfaces slide past each other, they resist movement due to friction. This resistance is known as kinetic friction when the objects are in motion. In this problem, the rock experiences kinetic friction as it slides along a rough surface. The force of kinetic friction can be calculated using the formula:

• \( f_k = \mu_k \cdot m \cdot g \)

where:

• \( \mu_k \) is the coefficient of kinetic friction, a measure of how rough or slippery the surface is.
• \( m \) is the mass of the object, in this case, the rock.
• \( g \) is the acceleration due to gravity, typically \( 9.81 \, \text{m/s}^2 \).

In this scenario, the coefficient of kinetic friction is 0.200, the mass of the rock is 20.0 kg, and gravity is acting downwards on the rock, all of which help us calculate the force stopping the rock.

Power Calculation

Power is the rate at which work is done or energy is transferred over time. In the context of this problem, it represents how quickly the frictional force stops the rock. To calculate the power produced by friction, use the formula:

• \( P = \frac{W}{t} \)

where:

• \( W \) is the work done by friction.
• \( t \) is the time it takes for the rock to stop.

The rock's kinetic energy is being transferred to thermal energy due to friction. Since energy is lost as the rock stops moving, the average power can be explicitly calculated to show how fast this energy conversion occurs. This is crucial in understanding how much energy is dissipated in a given time period.

Work Done by Friction

Work is defined as a force causing the movement—or a change in movement—of an object. For friction, work can be calculated as:

• \( W = f_k \cdot d \)

where:

• \( f_k \) is the force of kinetic friction.
• \( d \) is the distance over which the force acts.

However, since the rock starts with kinetic energy and comes to a stop, all this energy is converted into work done by friction. The work done can also be found using the formula:

• \( W = -\frac{1}{2} m v^2 \)

Here, \( v \) is the initial velocity of the rock as it begins to slide. The negative sign indicates that energy is being lost or dissipated as thermal energy and sound due to the frictional force.

Physics Problem-Solving

Solving physics problems systematically involves several steps. First, identify the forces at play, like the kinetic friction in this scenario. Calculate the frictional force using known coefficients and physical constants, such as gravity. Then, relate these forces to motion using principles like Newton’s laws.
Next, determine energy transformations; for instance, the conversion of kinetic energy to thermal energy due to friction. Equate the energy at the start to the work done by the stopping force to find distances or other unknown variables. Finally, calculate how the work and energy relate over time to find power.

• Identify known values and appropriate equations.
• Apply physics laws consistently to determine unknowns.
• Check results for physical plausibility.

A robust problem-solving strategy ensures that each step logically builds on the previous one, leading to a correct and understandable solution.