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Chapter 4: Problem 87

The alarm at a fire station rings and an 86-kg fireman, starting from rest, slides down a pole to the floor below (a distance of 4.0 m). Just before landing, his speed is 1.4 m/s. What is the magnitude of the kinetic frictional force exerted on the fireman as he slides down the pole?

Short Answer

Expert verified
The kinetic frictional force is 820.33 N.

Step by step solution

01

Identify the given data and required information

We need to determine the magnitude of the kinetic frictional force. The fireman's mass is 86 kg, the height from which he slides is 4.0 meters, and his speed just before landing is 1.4 m/s.
02

Calculate potential energy at the top

The gravitational potential energy (PE) at the top of the slide is given by the formula:\[ PE = mgh \]where \( m \) is mass (86 kg), \( g \) is acceleration due to gravity (9.8 m/s虏), and \( h \) is height (4.0 m).\[ PE = 86 \times 9.8 \times 4 = 3365.6 \text{ J} \]
03

Calculate kinetic energy just before landing

The kinetic energy (KE) just before landing can be calculated using the formula:\[ KE = \frac{1}{2}mv^2 \]where \( m = 86 \text{ kg} \) and \( v = 1.4 \text{ m/s} \).\[ KE = \frac{1}{2} \times 86 \times (1.4)^2 = 84.28 \text{ J} \]
04

Determine the work done by friction

Since the work done by friction (W_friction) is the difference in mechanical energy, we use the equation:\[ W_{friction} = PE - KE \]\[ W_{friction} = 3365.6 - 84.28 = 3281.32 \text{ J} \]
05

Calculate the frictional force

The work done by the frictional force is also given by:\[ W_{friction} = f \cdot d \]where \( f \) is the frictional force and \( d \) is the distance (4.0 m).\[ f = \frac{W_{friction}}{d} = \frac{3281.32}{4} = 820.33 \text{ N} \]
06

Conclusion

The magnitude of the kinetic frictional force exerted on the fireman is 820.33 N.

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Key Concepts

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

Potential Energy
Potential energy is the energy stored in an object due to its position relative to a gravitational field. For the fireman in the exercise, his potential energy depends on how high he is above the ground. This stored energy is important because it is what gets converted into other forms of energy when the fireman moves. Here's how it works:

  • The fireman's mass is 86 kg, and he slides from a height of 4 meters.
  • Using the formula for gravitational potential energy, \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is height, his potential energy is calculated as 3365.6 J.
  • This energy is initially what the fireman 'has' before he starts sliding down the pole.
As a general rule, the higher up an object is, the more potential energy it has. This energy plays a crucial role as it converts to kinetic energy when the object (or fireman) starts moving downwards, influenced by gravity.
Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. When the fireman slides down the pole, his potential energy gets converted into kinetic energy. This conversion allows us to see how fast and with how much force he is moving just before reaching the floor. Here's the breakdown:

  • The fireman, moving at a speed of 1.4 m/s before hitting the ground, has kinetic energy.
  • The formula for kinetic energy is \( KE = \frac{1}{2}mv^2 \), with \( m \) being mass and \( v \) being velocity.
  • In this scenario, his kinetic energy upon nearing the floor is calculated as 84.28 J.
Understanding kinetic energy is important because it gives insight into how much work has been done on the fireman by other forces, such as gravity and friction, as he descends.
Work-Energy Principle
The work-energy principle is a core concept in physics that relates the work done on an object to its change in energy. For the fireman, it explains how his initial potential energy is transformed and shared between kinetic energy and energy lost to friction.

  • The total mechanical energy at the top, 3365.6 J of potential energy, transforms as he slides down.
  • The kinetic energy of 84.28 J at the bottom represents only a part of his total energy.
  • The difference, 3281.32 J, accounts for the work done by the frictional force which effectively reduced his speed.
This principle highlights the conservation of energy, where total energy is transferred from one form to another, illustrating why energy is not lost but converted into forms like heat, due to friction in this case.
Frictional Force Calculation
The calculation of frictional force involves determining how much of the initial energy gets transformed into heat and other forms due to friction. In our fireman's descent, friction slows him down, and we can calculate it by understanding how that energy is used up.

  • Friction adds resistance to motion. In this case, it鈥檚 the kinetic friction as the fireman slides.
  • The work done by friction, formula-wise, is \( W_{friction} = f \cdot d \), showing how force and distance interact.
  • By knowing the work done by friction (3281.32 J) and the distance (4 m), the frictional force \( f \) is found to be 820.33 N.
This calculation reveals how much the slide slows down due to frictional forces, a critical value to ensure realistic expectations of motion and safety.

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