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Chapter 7: Problem 70

\(\bullet\) \(\bullet\) A 20.0 kg rock slides on a rough horizontal surface at 8.00 \(\mathrm{m} / \mathrm{s}\) and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is \(0.200 .\) What average thermal power is produced as the rock stops?

Short Answer

Expert verified
Average thermal power is calculated by using the work done by friction and the time it takes for the rock to stop.

Step by step solution

01

Identify the Given Variables

We are given the mass of the rock, \(m = 20.0 \, \text{kg}\), its initial velocity, \(v_i = 8.00 \, \text{m/s}\), and the coefficient of kinetic friction, \(\mu_k = 0.200\).
02

Find the Force of Kinetic Friction

The force of kinetic friction, \(F_k\), is calculated using the formula \(F_k = \mu_k \cdot m \cdot g\), where \(g = 9.8 \, \text{m/s}^2\) is the acceleration due to gravity. Substituting the values, \(F_k = 0.200 \times 20.0 \, \text{kg} \times 9.8 \, \text{m/s}^2\).
03

Calculate the Work Done by Friction

The work done by friction, \(W\), is equal to the change in kinetic energy of the rock. Since the rock stops, its final kinetic energy is zero and its initial kinetic energy is \(\frac{1}{2} m v_i^2\). Thus, \(W = 0 - \frac{1}{2} \times 20.0 \, \text{kg} \times (8.00 \, \text{m/s})^2\).
04

Determine the Thermal Power Produced

The thermal power is the rate at which work is done by the friction force, calculated by equation \(P = \frac{W}{t}\), where \(t\) is the time taken. First find the time using \(v_f = v_i - \mu_k \cdot g \cdot t\) with \(v_f = 0\). Rearrange and solve for \(t\), then use it to find \(P\).
05

Solve for Time

Calculate the stopping time using the equation \(v_f = 0 = 8.00 \, \text{m/s} - (0.200 \times 9.8 \, \text{m/s}^2 \times t)\). Solve this for \(t\) to find the stopping time of the rock.
06

Calculate Average Thermal Power

Use the work done \(W\) and calculated time \(t\) to calculate the average power \(P\) using \(P = \frac{|W|}{t}\). This is the thermal power generated by the friction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Power
Thermal power refers to the rate at which heat is generated due to friction between two surfaces. As the rock slides across the rough surface, the kinetic energy initially possessed by the rock is converted into heat energy due to the force of kinetic friction. This conversion of energy results in an increase in thermal energy, which is manifested as heat.
  • When a moving object like a rock slows down due to friction, the energy that it loses is expressed as thermal power.
  • This energy transformation occurs at a particular rate and is quantitatively represented by the formula: \[ P = \frac{W}{t} \]where \(P\) is the thermal power, \(W\) is the work done by friction, and \(t\) is the time taken for the object to come to a stop.
In the context of our exercise, this thermal power represents the average heat energy produced per unit time as the rock grinds to a halt due to friction.
Work-Energy Principle
The work-energy principle is a concept that helps to understand the changes in kinetic energy related to the work done on an object. According to this principle, the total work done by all the forces acting on an object equals the change in its kinetic energy.
  • For our rock, the work done by kinetic friction is responsible for the change from initial kinetic energy, when the rock is moving, to zero kinetic energy, when it stops.
  • This principle can be expressed mathematically as:\[ W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \]where \(v_f\) is the final velocity and \(v_i\) is the initial velocity. However, since \(v_f\) is zero (as the rock stops), the work done is equal to the initial kinetic energy multiplied by negative one.
Thus, the work-energy principle connects the dots between work done by friction and the subsequent energy loss, which eventually results in thermal power generation.
Frictional Force
Frictional force is a resistive force that acts in the opposite direction of movement. Kinetic friction occurs specifically when two surfaces slide against each other. This force plays a crucial role in stopping the rock and transforming its kinetic energy into thermal energy.
  • 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, \(m\) is the mass, and \(g\) is the acceleration due to gravity.
  • In our example, this friction causes the rock to decelerate, bringing it to a stop while simultaneously converting kinetic energy into heat.
Frictional force, while generally seen as energy-consuming, is also energy-transforming, as it facilitates the conversion of mechanical energy into thermal energy.
Stopping Time
Stopping time refers to the duration it takes for a moving object to come to rest due to opposing forces. In this exercise, the rock stops moving forward due to friction, and calculating this time is essential to determine the thermal power.
  • Using the equation for final velocity in uniformly decelerated motion, stopping time can be determined with:\[ t = \frac{v_i}{\mu_k \cdot g} \]This rearranged equation helps to solve for the time \(t\) when the final velocity \(v_f\) becomes zero.
  • In our problem, after working out the math, the stopping time gives a clear sense of how fast the kinetic energy is decreasing, thereby shedding light on the rate of thermal power generation per second.
Thus, stopping time is a vital part of quantifying the rate of energy transformation from kinetic to thermal power as friction works on the rock.

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Most popular questions from this chapter

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