/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 The weight of the block in the d... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 60

The weight of the block in the drawing is \(88.9 \mathrm{~N}\). The coefficient of static friction between the block and the vertical wall is \(0.560 .\) (a) What minimum force \(\vec{F}\) is required to prevent the block from sliding down the wall? (Hint: The static frictional force exerted on the block is directed upward, parallel to the wall.) (b) What minimum force is required to start the block moving up the wall? (Hint: The static frictional force is now directed down the wall.)

Short Answer

Expert verified
(a) 158.75 N; (b) 202.05 N.

Step by step solution

01

Understanding the Problem

We need to calculate force \( \overrightarrow{\mathrm{F}} \) required first to prevent the block from sliding down and second to move the block upward against the wall, given a weight and static friction coefficient.
02

Identify Forces Involved

The forces acting on the block along the wall surface are its weight acting downward \( W = 88.9 \mathrm{~N} \), the static friction \( f_s \), and the applied force \( F \) acting horizontally. Static friction opposing the block's motion will act differently based on the direction of motion.
03

Considering the Block Sliding Down

If the block is about to slide down, static friction \( f_s \) acts upward. The force of static friction is \( f_s = \mu_s N \), where \( \mu_s = 0.560 \) and \( N \) is the normal force, which is equal to \( F \) since it acts horizontally.
04

Equations for Preventing Slide Down

For equilibrium vertically (prevent sliding down), the static friction balances the weight. So, \( f_s = W \). We have:\[ \mu_s F = W \]\[ 0.560 F = 88.9 \]Solve for \( F \):\[ F = \frac{88.9}{0.560} \approx 158.75 \mathrm{~N} \]
05

Considering the Block Moving Up

When initiating upward motion, static friction \( f_s \) acts downward. Here, the applied force \( F \) must overcome both the weight of the block and the static friction.
06

Equations for Initiating Motion Upward

For initiating upward motion, the total force needs to satisfy:\[ F = W + f_s = W + \mu_s N \]Substitute \( N = F \):\[ F = 88.9 + 0.560 F \]Rearrange and solve for \( F \):\[ F - 0.560 F = 88.9 \]\[ 0.440 F = 88.9 \]\[ F = \frac{88.9}{0.440} \approx 202.05 \mathrm{~N} \]

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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 a kind of force that keeps an object at rest when it is placed against another surface. It's crucial to understand that static friction acts when the object is not moving. It works to prevent any movement until enough force is applied to overcome it. Imagine a block lying on a rough surface; the reason it doesn鈥檛 move when you push gently is because static friction holds it in place.
Here's how it works in this scenario:
  • Static friction acts in the opposite direction to the potential movement of the block.
  • This friction is dependent on the normal force, which comes from the contact between the surfaces.
  • The coefficient of static friction (\( \mu_s \)) determines how strong this force is between two surfaces.
To find the static frictional force, we use the equation: \[ f_s = \mu_s N \] where \( f_s \) is the static frictional force, \( \mu_s \) is the coefficient of static friction, and \( N \) is the normal force acting on the block.
Force Equilibrium
Force equilibrium is a critical concept that describes when all forces acting on an object are balanced, resulting in the object remaining in a constant state, either at rest or moving uniformly. When forces are in equilibrium, the object does not accelerate. This principle is used to determine the minimum force required to prevent the block from falling down or moving up.
In this exercise:
  • For the block not to slide down, the static friction must precisely counteract the gravitational pull (which is its weight).
  • For the block to just begin moving upwards, the applied force must overcome both gravity and the backward static friction force.
The logic here is simple: balance the forces. For preventing the downward slide: \[ \mu_s F = W \] For initiating upward motion: \[ F = W + \mu_s N \] Both equations revolve around balancing all pertinent forces acting on the block.
Normal Force
The normal force is a fundamental concept in physics. It is the perpendicular contact force exerted by a surface against an object resting on it. The normal force is crucial when calculating friction because the frictional force is proportional to it. Without this force, objects like the block in this exercise would simply sink through surfaces they rest upon!
Key points about the normal force:
  • Acts perpendicular to the contact surface.
  • In this problem, the normal force is the same as the force applied horizontally because the surface is vertical.
  • The value of normal force affects the magnitude of the static friction.
The normal force can be identified in formulas through \( N \), and is critical when calculating the static friction with the equation: \[ f_s = \mu_s N \]

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Most popular questions from this chapter

Jupiter is the largest planet in our solar system, having a mass and radius that are, respectively, 318 and 11.2 times that of earth. Suppose that an object falls from rest near the surface of each planet and that the acceleration due to gravity remains constant during the fall. Each object falls the same distance before striking the ground. Determine the ratio of the time of fall on Jupiter to that on earth.

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