Problem 65 - College Physics

Question

A 40 -kg crate is at rest on a level surface. If the coefficient of static friction between the crate and the surface is 0.69 , what horizontal force is required to get the crate moving?

Step-by-Step Solution

Answer

A horizontal force of 270.48 N is required to start moving the crate.

Step 1: Understanding the Force of Static Friction

The force of static friction is the resisting force that needs to be overcome to start moving the crate. It is calculated using the formula \( f_s = \mu_s N \), where \( \mu_s \) is the coefficient of static friction and \( N \) is the normal force acting on the object. For an object on a flat surface, the normal force \( N \) is equal to the gravitational force \( mg \).

Step 2: Calculating the Normal Force

The normal force \( N \) is equal to the gravitational force exerted on the crate. Given the mass \( m = 40 \) kg and the acceleration due to gravity \( g = 9.8 \) m/s², we calculate \( N = mg = 40 \times 9.8 = 392 \) N.

Step 3: Using the Static Friction Formula

Substitute the values of the coefficient of static friction \( \mu_s = 0.69 \) and the normal force \( N = 392 \) N into the static friction formula: \( f_s = \mu_s N = 0.69 \times 392 \).

Step 4: Calculating the Required Force

Perform the multiplication to find the force of static friction: \( f_s = 0.69 \times 392 = 270.48 \) N. This is the minimum horizontal force required to overcome static friction and initiate movement.

Key Concepts

Normal Force

In physics, the normal force is the support force exerted perpendicular to the surface on which an object rests. This force balances out the gravitational force pulling the object downwards.
In the case of a crate resting on a level surface, the normal force can be calculated using the formula:

• For an object on a level surface, the normal force ( N ) equals the gravitational force ( mg ).
• This is derived from N = mg , where m is the mass and g is the acceleration due to gravity.

For our example with a 40-kg crate, the normal force is calculated as 392 Newtons (N), assuming gravity is 9.8 m/s². So, the normal force is essential in defining the contact interaction between the object and the surface.

Coefficient of Static Friction

The coefficient of static friction, denoted as μ_s , is a dimensionless value that represents the frictional "stickiness" between two surfaces.
It's a measure of how difficult it is to get an object moving from rest. Here's what to keep in mind:

• μ_s varies depending on the materials involved.
• In this exercise, the coefficient is 0.69, indicating a reasonably high static friction power between the crate and the surface.

The static friction force, which needs to be overcome to move the crate, is calculated using f_s = μ_s N , where N is the normal force. A higher μ_s means more force is required to initiate movement. The interaction force, influenced by both normal force and static friction coefficient, crucially determines movement thresholds.

Gravitational Force

Gravitational force is the force of attraction between any two masses, like the Earth and the crate. This force can be computed using the formula:

• F_{ ext{gravity}} = mg , where m is the mass and g is the gravitational acceleration (approximately 9.8 m/s² on Earth).
• It's responsible for keeping objects grounded.

In this scenario, the gravitational force acting on our 40-kg crate is 392 N. Understanding gravity's role is fundamental in force calculations because it directly affects the normal force and subsequently the static friction force as well. Every weight on a surface generates an equal and opposite force, which is crucial in aligning calculations for static and dynamic interactions.

Force Calculation

To get an object to move, you must overcome static friction. Force calculation revolves around finding that minimum amount of force. In our specific exercise, this involves:

• Using the static friction formula f_s = μ_s N .
• With μ_s = 0.69 and N = 392 N , we determine the required force as 270.48 N.

Thus, a force of at least 270.48 N is needed to start moving the crate.
By applying the right amount of horizontal force, you can surpass static friction, transforming the object's state from rest to motion. This calculation is a prime example of how physics principles are applied to solve real-world problems.