Problem 2 - College Physics

Question

A \(5.0-\mathrm{kg}\) box slides a 10 -m distance on ice. If the coefficient of kinetic friction is 0.20 , what is the work done by the friction force?

Step-by-Step Solution

Answer

The work done by the friction force is \(-98 \, \text{J}\).

Step 1: Identify the Forces

The only force doing work against the box's motion is the kinetic friction force. We know the coefficient of kinetic friction \( \mu_k \) is 0.20.

Step 2: Calculate Normal Force

The normal force \( F_n \) for an object on a flat surface is equal to the gravitational force on the object. The gravitational force is the weight, \( F_n = mg \), where \( m = 5.0 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). Thus, \( F_n = 5.0 \times 9.8 = 49 \, \text{N} \).

Step 3: Calculate Frictional Force

The frictional force \( f_k \) is given by the product of the coefficient of kinetic friction and the normal force: \( f_k = \mu_k F_n \). Substituting the known values, \( f_k = 0.20 \times 49 = 9.8 \, \text{N} \).

Step 4: Calculate Work Done by Friction

The work done by the friction force is given by \( W = f_k \times d \times \cos(\theta) \), where \( \theta \) is the angle between the force and the direction of motion. For frictional force, \( \theta = 180^\circ \) because it opposes the motion. Therefore, \( \cos(180^\circ) = -1 \). Substituting the values, \( W = 9.8 \times 10 \times (-1) = -98 \, \text{J} \).

Key Concepts

Kinetic Friction Explained

When an object slides over a surface, it experiences a force opposing its motion. This force is known as kinetic friction.
This type of friction occurs due to the interaction between the contact surfaces.
It always acts in the opposite direction to the movement of the object, aiming to slow it down.Some important points to consider with kinetic friction:

• Kinetic friction depends on the nature of the surfaces in contact and the normal force.
• The coefficient of kinetic friction (\( \mu_k \)) is a dimensionless constant, varying between different material pairs.
• Higher values of \( \mu_k \) indicate more frictional resistance.

In our problem, the coefficient is given as 0.20, which points to relatively low friction, typical for ice.
This means the box on ice would experience moderate sliding resistance.

Understanding the Normal Force

The normal force is a supporting force exerted by a surface perpendicular to the object resting on it.
In simple terms, it counteracts the weight of the object pressing down.
For a box resting on a flat surface, the normal force matches the gravitational force pulling it down. Here are key points about normal force:

• It acts perpendicular to the surface.
• On a horizontal surface, the normal force equals the object’s weight.
• If the surface is inclined, calculations involve vector components to determine the correct value.

In the exercise, the normal force was calculated as \( 49 \, \text{N} \), using the weight formula \( F_n = mg \), where \( m = 5.0 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).

What is Frictional Force?

Frictional force is the resistive force that prevents two surfaces from sliding smoothly past each other.
In the case of movement, it's kinetic friction involved.
Frictional force plays a crucial role in controlling the motion of objects.Key characteristics of frictional force:

• It depends on the nature of surfaces in contact and the normal force.
• Frictional force can be static or kinetic, based on whether the object is stationary or moving.
• In many calculations, frictional force is considered as \( f_k = \mu_k \cdot F_n \) for kinetic friction.

In our scenario, the frictional force opposing the box’s movement was calculated as \( 9.8 \, \text{N} \).
This value was derived by multiplying the coefficient of kinetic friction (\( 0.20 \)) with the normal force (\( 49 \, \text{N} \)).
This frictional force then helps calculate the work done by friction against the slide, which amounts to \( -98 \, \text{J} \), reflecting the energy lost in overcoming friction.