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?

Verified step by step guidance

1. Identify the forces acting on the rock. The primary force doing work here is the force of kinetic friction, which opposes the motion of the rock.
2. Calculate the force of kinetic friction using the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>f</mi><mo>=</mo><mi>&#956;</mi><msub><mi>F</mi><mi>n</mi></msub></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>&#956;</mi></math> is the coefficient of kinetic friction and <math xmlns='http://www.w3.org/1998/Math/MathML'><msub><mi>F</mi><mi>n</mi></msub></math> is the normal force. Since the surface is horizontal, the normal force is equal to the gravitational force: <math xmlns='http://www.w3.org/1998/Math/MathML'><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><mi>m</mi><mi>g</mi></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>m</mi></math> is the mass of the rock and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>g</mi></math> is the acceleration due to gravity.
3. Determine the work done by friction as the rock comes to a stop. The work done by friction is given by: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>W</mi><mo>=</mo><mi>f</mi><mi>d</mi></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>d</mi></math> is the distance over which the force acts. Use the work-energy principle, which states that the work done by friction is equal to the change in kinetic energy: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>W</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>v</mi></math> is the initial velocity of the rock.
4. Calculate the time taken for the rock to stop using the equation: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>t</mi><mo>=</mo><mfrac><mi>v</mi><mi>a</mi></mfrac></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>a</mi></math> is the acceleration. The acceleration can be found using Newton's second law: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>a</mi><mo>=</mo><mfrac><mi>f</mi><mi>m</mi></mfrac></math>.
5. Finally, calculate the average power produced by friction using the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>P</mi><mo>=</mo><mfrac><mi>W</mi><mi>t</mi></mfrac></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>W</mi></math> is the work done by friction and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>t</mi></math> is the time taken for the rock to stop.

Key Concepts

Kinetic Friction

Kinetic friction is the force that opposes the motion of two surfaces sliding past each other. It is calculated using the formula F_k = μ_k * N, where μ_k is the coefficient of kinetic friction and N is the normal force. In this scenario, the normal force is equal to the gravitational force on the rock, which is its weight (mass times gravity).

Work-Energy Principle

The work-energy principle states that the work done by all forces acting on an object equals the change in its kinetic energy. As the rock slides to a stop, the work done by friction is equal to the initial kinetic energy of the rock. This principle helps in calculating the distance over which the friction acts to stop the rock.

Power

Power is the rate at which work is done or energy is transferred over time. It is calculated as P = W/t, where W is the work done and t is the time taken. In this problem, the average power produced by friction can be found by dividing the work done by friction (which equals the initial kinetic energy) by the time it takes for the rock to stop.