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Chapter 4: Problem 35

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?

Short Answer

Expert verified
Each hand must exert a force of 77.5 N.

Step by step solution

01

Understand the Forces Involved

The problem involves the weight of the book acting downwards and the forces from the hands acting horizontally, pressing the book from both sides. The static friction between the hands and the book provides the upward force required to keep the book from falling.
02

Identify the Force of Friction Required

The book's weight is the force that needs to be countered by friction. Therefore, the force of friction required is equal to the weight of the book, which is 31 N.
03

Use the Formula for Static Friction

The formula for static friction is given by: \[F_{f} = ext{friction force} = ext{coefficient of static friction} \times ext{normal force}\]Here, the friction force \( F_{f} \) must equal the book's weight: 31 N = 0.40 \( F_{N} \), where \( F_{N} \) is the normal force from each hand.
04

Calculate the Normal Force from Each Hand

Since the normal force from both hands must collectively counteract the weight, divide the equation by 0.40 to find the normal force exerted by one hand: \[F_{N} = \frac{31 \, ext{N}}{0.40} = 77.5 \, ext{N}\]
05

Conclude with Required Pressing Force Per Hand

Therefore, each hand must exert a force of 77.5 N to ensure that the book remains stationary and does not fall.

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Key Concepts

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

Normal Force
In physics, the "normal force" is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface. In our exercise, the normal forces are the ones applied by each hand pressing horizontally on the book.

These forces need to counteract the weight of the book by supporting it horizontally. Since the weight pulls the book downward, the normal force in each hand must be sufficient to generate an upward static friction force that counteracts this weight. This means that while it might seem like these forces only push on the book sides, they actually play a crucial role in keeping the book from falling by creating friction.
  • The normal force is perpendicular to the force of gravity.
  • In this scenario, it is responsible for generating adequate friction to prevent motion.
  • It needs to be enough to cause a frictional force greater than or equal to the weight of the book.
Coefficient of Friction
The coefficient of static friction is a measure that describes the ratio between the maximum static frictional force and the normal force. In our example, the coefficient is 0.40, indicating how "sticky" the surfaces in contact (hands and book) are.

The static friction force increases with the applied normal force until it reaches a threshold, determined by this coefficient. If the applied force goes over this threshold, the object would start moving.
  • Symbolized by \( \mu_s \), the coefficient is unitless.
  • A higher coefficient means more friction and better grip between the surfaces.
  • The formula linking friction and normal force is \( F_f = \mu_s \times F_N \).
Understanding this coefficient helps us calculate how much force we need to apply to prevent the book from slipping.
Forces and Motion
When analyzing "forces and motion", it is essential to understand how various forces come together to cause or inhibit movement. In our case, we want to keep the book stationary against the force of gravity pulling it downward.

Forces involved include:
  • Normal Force: Horizontal forces exerted by each hand.
  • Frictional Force: Opposes gravity, acts upward to prevent falling.
  • Gravitational Force: Weight of the book, acting downward.
The balance between these forces determines the book's state. By adjusting the normal forces, we directly influence the magnitude of the frictional force. Thus, for the book to remain still, the total upward friction must match the downward weight due to gravity.
Weight of an Object
The weight of an object, expressed in newtons (N), is the force due to gravity acting on its mass. In this scenario, the weight of the book is 31 N, meaning gravity exerts a force of 31 N downwards.

This weight must be counteracted by an equivalent amount of static friction to ensure that the book does not fall.
  • Weight is calculated as the product of mass and the acceleration due to gravity (\( W = mg \)).
  • It is always directed towards the center of the Earth.
  • Ensuring friction exceeds or matches this weight keeps the object stationary.
In our case, understanding the weight helps configure the necessary frictional force for balance, utilizing the normal force applied by hands to provide the required opposition.

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Most popular questions from this chapter

At a time when mining asteroids has become feasible, astronauts have connected a line between their 3500 -kg space tug and a 6200 -kg asteroid. Using their ship's engine, they pull on the asteroid with a force of \(490 \mathrm{~N}\). Initially the tug and the asteroid are at rest, \(450 \mathrm{~m}\) apart. How much time does it take for the ship and the asteroid to meet?

An arrow, starting from rest, leaves the bow with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\). If the average force exerted on the arrow by the bow were doubled, all else remaining the same, with what speed would the arrow leave the bow?

A student is skateboarding down a ram p that is \(6.0 \mathrm{~m}\) long and inclined at \(18^{\circ}\) with respect to the horizontal. The initial speed of the skateboarder at the top of the ramp is \(2.6 \mathrm{~m} / \mathrm{s}\). Neglect friction and find the speed at the bottom of the ramp.

A person is attempting to push a refrigerator across a room. He exerts a horizontal force on the refrigerator, but it does not move. (a) What other horizontal force must be acting on the refrigerator? (b) How are the magnitude and direction of this force related to the force that the person exerts? (c) Suppose that the person applies the force such that it is the largest possible force before the refrigerator begins to move. What factors determine the magnitude of this force? Problem Consult Multiple-Concept Example 9 to explore a model for solving this problem. A person pushes on a 57 -kg refrigerator with a horizontal force of \(-267 \mathrm{~N}\); the minus sign indicates that the force is directed along the \(-x\) direction. The coefficient of static friction is \(0.65 .\) (a) If the refrigerator does not move, what is the magnitude and direction of the static frictional force that the floor exerts on the refrigerator? (b) What is the magnitude of the largest pushing force that can be applied to the refrigerator before it just begins to move?

A 20.0 -kg sled is being pulled across a horizontal surface at a constant velocity. The pulling force has a magnitude of \(80.0 \mathrm{~N}\) and is directed at an angle of \(30.0^{\circ}\) above the horizontal. Determine the coefficient of kinetic friction.
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