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Chapter 4: Problem 39

A \(60.0-\mathrm{kg}\) crate rests on a level floor at a shipping dock. The coefficients of static and kinetic friction are 0.760 and 0.410 , respectively. What horizontal pushing force is required to (a) just start the crate moving and (b) slide the crate across the dock at a constant speed?

Short Answer

Expert verified
(a) 447.34 N to start moving; (b) 241.33 N to move at a constant speed.

Step by step solution

01

Determine the Normal Force

First, calculate the normal force (\(F_n\)) acting on the crate. Since the crate is on a level floor and there are no vertical forces besides gravity, the normal force is equal to the gravitational force on the crate. This is given by:\[F_n = mg\]where \(m = 60.0\,\text{kg}\) is the mass of the crate and \(g = 9.81\,\text{m/s}^2\) is the acceleration due to gravity. Substituting the values gives:\[F_n = 60.0 \times 9.81 = 588.6\,\text{N}.\]
02

Calculate the Force to Overcome Static Friction

To just start the crate moving, the pushing force must overcome the static friction. The maximum static frictional force \(f_{s, \text{max}}\) is calculated using the coefficient of static friction \(\mu_s = 0.760\):\[f_{s, \text{max}} = \mu_s F_n = 0.760 \times 588.6 = 447.336\,\text{N}.\]Therefore, a force of approximately \(447.34\,\text{N}\) is required to start moving the crate.
03

Calculate the Force to Overcome Kinetic Friction

Once the crate is moving, the force required to keep it moving at a constant speed is equal to the kinetic frictional force. The force of kinetic friction \(f_k\) is calculated using the coefficient of kinetic friction \(\mu_k = 0.410\):\[f_k = \mu_k F_n = 0.410 \times 588.6 = 241.326\,\text{N}.\]Thus, a force of approximately \(241.33\,\text{N}\) is needed to slide the crate at constant speed.

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Key Concepts

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

Static Friction
Static friction is the force that keeps an object at rest when it is subjected to external forces. This frictional force needs to be overcome to start moving the object. It acts in the opposite direction to the applied force attempting to move the object. The maximum static frictional force is the highest amount of static friction that can develop before the object begins to move.
In our scenario with the crate, the static friction can be calculated through the formula:
  • \( f_{s, \text{max}} = \mu_s F_n \)
where \( f_{s, \text{max}} \) is the maximum static frictional force, \( \mu_s \) is the coefficient of static friction, and \( F_n \) is the normal force. For the crate, this becomes 447.336 N, meaning any pushing force greater than this will initiate movement. Remember, the static frictional force doesn't have a fixed value because it adjusts to equal the applied force until reaching its maximum at \( f_{s, \text{max}} \).
Kinetic Friction
Kinetic friction comes into play once the object starts moving. Unlike static friction, kinetic friction remains constant, regardless of whether the speed of the object changes.
This is because kinetic friction relates to the force opposing the movement of two surfaces sliding past each other. To keep our crate moving at a steady pace, we need to maintain a force that balances out the kinetic friction. The formula to determine kinetic friction is:
  • \( f_k = \mu_k F_n \)
where \( f_k \) represents the kinetic friction, and \( \mu_k \) is the coefficient of kinetic friction. In the crate's situation, this force is calculated to be 241.326 N. Applying this force consistently will allow the crate to slide at a constant speed. While static friction handles starting motion, kinetic friction deals with maintaining movement.
Normal Force
Normal force is the support force exerted by a surface perpendicular to the object resting upon it. It acts in the opposite direction of gravity, effectively balancing the weight of an object resting on a surface.
For a level surface like the shipping dock, the normal force on the crate can be computed using the equation:
  • \( F_n = mg \)
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity. For the crate, with a mass of 60.0 kg, the normal force is 588.6 N. In scenario calculations, knowing the normal force is essential as it is directly involved in calculating both static and kinetic frictions. Understanding the normal force is crucial for predicting how forces like friction behave when we interact with objects on flat surfaces.

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Most popular questions from this chapter

A woman stands on a scale in a moving elevator. Her mass is 60.0 kg, and the combined mass of the elevator and scale is an additional 815 kg. Starting from rest, the elevator accelerates upward. During the acceleration, the hoisting cable applies a force of \(9410 \mathrm{N}\). What does the scale read during the acceleration?

A person is trying to judge whether a picture (mass \(=1.10 \mathrm{kg}\) ) is properly positioned by temporarily pressing it against a wall. The pressing force is perpendicular to the wall. The coefficient of static friction between the picture and the wall is \(0.660 .\) What is the minimum amount of pressing force that must be used?

A \(1.14 \times 10^{4}\) -kg lunar landing craft is about to touch down on the surface of the moon, where the acceleration due to gravity is \(1.60 \mathrm{m} / \mathrm{s}^{2}\). At an altitude of \(165 \mathrm{m}\) the craft's downward velocity is \(18.0 \mathrm{m} / \mathrm{s}\). To slow down the craft, a retrorocket is firing to provide an upward thrust. Assuming the descent is vertical, find the magnitude of the thrust needed to reduce the velocity to zero at the instant when the craft touches the lunar surface.

A student presses a book between his hands, as the drawing indicates. The forces that he exerts on the front and back covers of the book are perpendicular to the book and are horizontal. The book weighs \(31 \mathrm{N}\). The coefficient of static friction between his hands and the book is \(0.40 .\) To keep the book from falling, what is the magnitude of the minimum pressing force that each hand must exert?

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