Question

A box with mass 10.0 kg moves on a ramp that is inclined at an angle of 55.0\(^\circ\) above the horizontal. The coefficient of kinetic friction between the box and the ramp surface is \(\mu_k =\) 0.300. Calculate the magnitude of the acceleration of the box if you push on the box with a constant force \(F =\) 120.0 N that is parallel to the ramp surface and (a) directed down the ramp, moving the box down the ramp; (b) directed up the ramp, moving the box up the ramp.

Short answer

Down the ramp: 18.33 m/s²; Up the ramp: 2.294 m/s².

Step-by-step solution

Identify Forces and Components

The forces acting on the box include gravitational force, applied force, frictional force, and normal force. To analyze these forces, it's important to resolve the gravitational force into components parallel and perpendicular to the ramp. The parallel component is given by \( mg \sin(55^\circ) \) and the perpendicular component is \( mg \cos(55^\circ) \). Here, \( m = 10.0 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \).

Calculate Gravitational Force Components

First, calculate the components of the gravitational force:- Perpendicular to the ramp: \[ F_{\text{gravity-perpendicular}} = mg\cos(\theta) = 10.0 \times 9.8 \times \cos(55^\circ) \approx 56.2 \text{ N} \]- Parallel to the ramp: \[ F_{\text{gravity-parallel}} = mg\sin(\theta) = 10.0 \times 9.8 \times \sin(55^\circ) \approx 80.2 \text{ N} \]

Determine Frictional Force

The frictional force can be calculated using the normal force, which equals the perpendicular component of the gravitational force for an incline without vertical acceleration. \[ F_{\text{friction}} = \mu_k F_{\text{gravity-perpendicular}} = 0.300 \times 56.2 = 16.86 \text{ N} \]

Calculate Net Force in Case (a) Down the Ramp

When the box is pushed down the ramp, the net force is a combination of the applied force, the parallel component of gravity, and the frictional force acting in the opposite direction.\[ F_{\text{net}} = F + F_{\text{gravity-parallel}} - F_{\text{friction}} \]\[ F_{\text{net}} = 120.0 + 80.2 - 16.86 = 183.34 \text{ N} \]

Calculate Acceleration Down the Ramp

Using Newton's second law, calculate the acceleration: \[ a = \frac{F_{\text{net}}}{m} = \frac{183.34}{10.0} = 18.33 \text{ m/s}^2 \]

Calculate Net Force in Case (b) Up the Ramp

When the box is pushed up the ramp, the net force is the difference between the applied force and the sum of the gravitational component parallel to the ramp and the frictional force.\[ F_{\text{net}} = F - F_{\text{gravity-parallel}} - F_{\text{friction}} \]\[ F_{\text{net}} = 120.0 - 80.2 - 16.86 = 22.94 \text{ N} \]

Calculate Acceleration Up the Ramp

Using Newton's second law, calculate the acceleration:\[ a = \frac{F_{\text{net}}}{m} = \frac{22.94}{10.0} = 2.294 \text{ m/s}^2 \]

Key concepts

Friction on an Inclined Plane

Friction is a force that opposes motion between two surfaces in contact. When we deal with objects on an inclined plane, friction plays a crucial role in determining motion. On a ramp, the frictional force is affected by the angle of the incline and the nature of the surfaces in contact. The frictional force can be calculated using the formula:
\[ F_{\text{friction}} = \mu_k F_{\text{normal}} \]
where

• \( F_{\text{friction}} \) is the frictional force,
• \( \mu_k \) is the coefficient of kinetic friction, and
• \( F_{\text{normal}} \) is the normal force.

On an inclined plane, the normal force is smaller than when the object is on a flat surface because it is equal to the perpendicular component of the gravitational force. Understanding how friction works on an inclined plane is essential for solving problems involving motion and acceleration, as seen in the exercise with the box on the ramp. Remember, the friction force opposes the motion, regardless if the box is moving up or down the incline.

Gravitational Force Components

Gravitational force is the force of attraction between an object and the Earth. When analyzing objects on an inclined plane, it's important to break down the gravitational force into two components:

• Parallel component: \( F_{\text{gravity-parallel}} = mg\sin(\theta) \)
• Perpendicular component: \( F_{\text{gravity-perpendicular}} = mg\cos(\theta) \)

In these equations,

• \( m \) represents the mass of the object,
• \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)), and
• \( \theta \) is the angle of the incline.

The parallel component pulls the object down along the plane, contributing to the motion, while the perpendicular component decides the normal force, which affects the frictional force. Fully grasping these components is vital for understanding the forces acting on an object on a slope like the box in the ramp problem.

Kinetic Friction Coefficient

The kinetic friction coefficient, \( \mu_k \), is a dimensionless value that characterizes the friction between moving surfaces. It varies based on materials and conditions but remains constant for specific surfaces. In the context of an inclined plane:

• \( \mu_k \) determines the magnitude of the frictional force.
• The formula \( F_{\text{friction}} = \mu_k F_{\text{normal}} \) shows how the frictional force relates to the normal force.
• Common surfaces can have a kinetic friction coefficient anywhere from 0 to 1, or even higher with very sticky surfaces.

In the solved exercise, the kinetic friction coefficient of 0.300 indicates moderate friction, typical of many surface interactions. Understanding \( \mu_k \) helps you to predict the resistance an object will face when moving, especially when pushing motions involve surfaces with friction, as illustrated in the box scenario on a ramp.