Problem 66 - University Physics with Modern Physics

Question

A 6.00 -kg box sits on a ramp that is inclined at \(37.0^{\circ}\) above the horizontal. The coefficient of kinetic friction between the box and the ramp is \(\mu_{k}=0.30 .\) What horizontal force is required to move the box up the incline with a constant acceleration of 4.20 \(\mathrm{m} / \mathrm{s}^{2}\) ?

Step-by-Step Solution

Answer

The horizontal force required is approximately 61.2 N.

Step 1: Determine the Components of Gravity

First, we need to decompose the gravitational force acting on the box into components parallel and perpendicular to the inclined plane. The gravitational force can be calculated as \( F_g = mg \) where \( m = 6.00 \text{ kg} \) and \( g = 9.81 \text{ m/s}^2 \). The component along the ramp is \( F_{g, \, parallel} = mg \sin(\theta) \) and the component perpendicular to the ramp is \( F_{g, \, perpendicular} = mg \cos(\theta) \).

Step 2: Calculate the Frictional Force

Using the component of gravity perpendicular to the incline, calculate the normal force since there is no vertical movement. Therefore, \( N = F_{g, \, perpendicular} = mg \cos(\theta) \). The frictional force \( f_k \) is then \( \mu_k N \).

Step 3: Sum Forces Along the Incline

Applying Newton's second law along the ramp, the net force \( F_{net} \) required to achieve the acceleration of \( 4.20 \text{ m/s}^2 \) is given by \( F_{net} = ma = F_{applied} - F_{g, \, parallel} - f_k \), where \( F_{applied} \) is the component of the horizontal force along the ramp.

Step 4: Relate Horizontal Force to Applied Force

The horizontal force \( F_h \) must be expressed in terms of its component along the incline. Since the ramp is inclined at \( 37.0^{\circ} \), \( F_{applied} = F_h \cos(\theta) \). Substitute into the equation from the previous step.

Step 5: Solve for the Horizontal Force

Rearrange the equation from the previous step to solve for the horizontal force \( F_h \): \( F_h = \frac{ma + mg\sin(\theta) + f_k}{\cos(\theta)} \). Insert the values for each parameter: mass, gravity, friction coefficient, angle, and desired acceleration.

Key Concepts

Newton's Second Law

Newton's Second Law is a foundational principle in physics that describes how the motion of an object changes when it is subjected to external forces. According to the law, the acceleration of an object depends on two variables: the net force acting upon the object and its mass. The law is typically expressed with the equation:

• \( F_{net} = ma \)

In this equation, \( F_{net} \) is the net force measured in newtons (N), \( m \) is the mass in kilograms (kg) and \( a \) is the acceleration in meters per second squared (m/s²).
When dealing with objects on inclined planes, like the box in the problem, Newton's Second Law helps us determine the necessary force to achieve a specific acceleration. The net force will be a combination of the applied horizontal force, gravitational force, and frictional force acting on the box.
By calculating the net force required, students can find the necessary horizontal force which when applied, controls the box’s motion on the ramp.

Inclined Plane

An inclined plane, or ramp, is a flat surface tilted at an angle to the horizontal. It serves as a simple machine making it easier to raise objects to higher heights. The angle of inclination affects how forces need to be calculated and applied. For our problem, the ramp is inclined at \( 37.0^{\circ} \).
The important forces to consider on an inclined plane include:

• The gravitational force pulls the object downward.
• Perpendicular (normal) force acts at a right angle to the plane.
• Frictional force opposes the motion and acts along the plane.

When the box on a ramp needs to move upwards, like in the given problem, understanding these forces becomes crucial. The direction and magnitude of the forces determine the amount of additional force needed to reach the desired acceleration. Recognizing and calculating these forces for different angles and surfaces is essential for solving physics problems involving inclined planes.

Kinetic Friction

Kinetic friction is the resisting force that occurs when two surfaces move against each other. It plays a significant role in determining the amount of force required to move the box along the inclined plane. In our problem, the coefficient of kinetic friction \( \mu_k \) between the box and the ramp is given as \( 0.30 \).
The frictional force can be calculated using the formula:

• \( f_k = \mu_k N \)

where \( f_k \) is the frictional force and \( N \) is the normal force. The normal force comes from the gravitational component perpendicular to the inclined plane. That’s why, despite the box moving parallel to the incline, it's essential to calculate the perpendicular force component to determine friction.
Kinetic friction always acts in the opposite direction of the movement, so when you apply additional force to move the box up the plane, the friction will oppose it, requiring even more force.

Gravitational Force Decomposition

Gravitational force decomposition is a method used to simplify the analysis of motion on inclined planes. Instead of dealing with the gravitational force as a whole, it is broken down into components parallel and perpendicular to the plane. This decomposition helps us understand how much of the gravitational force contributes to downward motion and how much acts as a normal force.
For an inclined plane at an angle \( \theta \), the formulas are:

• Parallel component: \( F_{g, \, parallel} = mg \sin(\theta) \)
• Perpendicular component: \( F_{g, \, perpendicular} = mg \cos(\theta) \)

These components are crucial. The parallel component pulls the box down the incline, while the perpendicular component influences the normal force used in friction calculations. Understanding gravitational force decomposition enables us to tackle and resolve the forces an object experiences on a slope effectively.