Question

An \(800-\mathrm{N}\) box is pushed up an inclined plane that is \(4.0 \mathrm{~m}\) long. It requires \(3200 \mathrm{~J}\) of work to get the box to the top of the plane, which is \(2.0 \mathrm{~m}\) above the base. What is the magnitude of the average friction force on the box? (Assume the box starts at rest and ends at rest.) a) zero b) not zero but less than \(400 \mathrm{~N}\) c) greater than \(400 \mathrm{~N}\) d) \(400 \mathrm{~N}\) e) \(800 \mathrm{~N}\)

Short answer

Answer: The magnitude of the average friction force on the box is greater than 400 N.

Step-by-step solution

1. Determine the gravitational force on the box

To calculate the gravitational force acting on the box, simply multiply the weight by the gravitational acceleration, g (9.81 m/s^2). This gives us: Force = Weight × g = 800 N × 9.81 m/s² = 7848 N

2. Calculate the work done against gravity

Next, we need to calculate the work done against gravity. This work is equal to the gravitational force multiplied by the height the box is lifted: Work against gravity = Force × height = 7848 N × 2.0 m = 15,696 J

3. Determine the work done against friction

Now subtract the work done against gravity from the total work required to lift the box. This will give us the work done against friction: Work against friction = Total work - Work against gravity = 3200 J - 15,696 J = -12,496 J The negative sign indicates that friction is acting in the opposite direction of motion. Since we are looking for the magnitude of the friction force, we can ignore the negative sign.

4. Calculate the distance traveled along the plane

We are given the length of the inclined plane as 4.0 m.

5. Determine the average friction force

To find the average friction force, divide the work done against friction by the distance traveled along the plane: Average friction force = Work against friction / Distance traveled = 12,496 J / 4.0 m = 3124 N This value indicates that the magnitude of the average friction force on the box is greater than 400 N. Thus, the correct answer is c) greater than 400 N.

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction that attracts any two objects with mass. In physics, this force is essential for understanding movements on Earth and throughout the cosmos. When dealing with problems on Earth, the gravitational force, often termed the weight of an object, is calculated as the mass of the object multiplied by the acceleration due to Earth's gravity, which is approximately 9.81 m/s².

For an object on an incline, such as a box on a ramp, gravitational force is what pulls the object straight down towards the Earth, making it harder to move the object upwards. The component of the gravitational force acting along the ramp is responsible for some of the work that needs to be done to move the object to the top. Understanding this force is critical when solving physics problems involving motion and energy, as you must know how much work must be done against gravity to move an object.

Work and Energy

The concepts of work and energy are closely related in physics. Work is defined as the effort needed to move an object over a distance when a force is applied, and it is given by the formula Work = Force x Distance. Energy, on the other hand, is the capacity to do work.

When solving problems involving work and energy, it's important to consider all forms of work being done, such as against gravity, friction, or any other forces present. The total work done on an object is equal to the change in its energy. In the case of lifting the box up an inclined plane, the energy input (or work done) is not only used to overcome gravitational force but also to overcome any opposing forces like friction, which leads to the concept of work done against friction.

Friction Force

Friction is the force that opposes the relative motion between two surfaces in contact. It is this resistance that we often need to overcome to move objects. Frictional forces can be helpful, like when it prevents slipping, or a challenge when we want to move an object smoothly.

In the context of the given problem, where a box is pushed up an inclined plane, friction is the force that acts in the opposite direction to the movement. To compute the average friction force, one must understand that the work done against friction, in combination with the work done against gravity, totals the energy expended to move the box. Thus, by calculating the work needed to overcome friction, it is possible to determine the magnitude of the friction force. Grasping this concept is critical when predicting motion and analyzing energy requirements for different scenarios.