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) \(0 \mathrm{~N}\) c) greater than \(400 \mathrm{~N}\) b) not zero but d) \(400 \mathrm{~N}\) less than \(400 \mathrm{~N}\) e) \(800 \mathrm{~N}\)

Short answer

Answer: The magnitude of the average frictional force acting on the box is approximately 309.2 N.

Step-by-step solution

Calculate the applied force using the work-energy principle

To find the applied force, we can use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy. Since the box starts and ends at rest, the change in kinetic energy is zero, so the work done on the box is in overcoming the weight force and friction force. We can write: \(W = F_{app}d \cdot \cos(\theta)\) where \(W\) is the work done (3200 J), \(F_{app}\) is the applied force we need to find, \(d\) is the length of the inclined plane (4.0 m), and \(\theta\) is the angle between the force vector and the displacement vector. First, calculate the angle between the inclined plane and the horizontal: \(\sin(\theta) = \frac{height}{length} = \frac{2.0\,\text{m}}{4.0\,\text{m}}\) \(\theta = \arcsin(0.5)\) Now, we can find the applied force: \(F_{app} = \frac{W}{d\cos(\theta)}\)

Calculate the components of gravity force and applied force

We need to determine the components of the applied force and gravitational force parallel to the inclined plane: \(F_{app,\parallel} = F_{app}\sin(\theta)\) \(F_{g,\parallel} = mg\sin(\theta)\) where \(m\) is the mass of the box, and \(g\) is the acceleration due to gravity (9.81 m/s²). Note that the weight of the box is \(800\,\text{N}\), so \(m = \frac{800\,\text{N}}{9.81\,\text{m/s²}}\).

Calculate the frictional force

To find the friction force, we can use the fact that the box starts and ends at rest, meaning that the net force acting on the box is zero. Thus, the frictional force should balance the other forces acting on the box in the direction parallel to the inclined plane: \(F_{friction} = F_{app,\parallel} - F_{g,\parallel}\) Substitute the calculated values of \(F_{app,\parallel}\) and \(F_{g,\parallel}\) and solve for \(F_{friction}\). The resulting value of \(F_{friction}\) will allow us to choose the correct answer from the given options.

Key concepts

Understanding the Work-Energy Principle

The work-energy principle in physics is a fundamental concept that states that the work done by forces on an object results in a change in its kinetic energy. When applying this principle, it's important to distinguish between the different kinds of work that might be done - for example, against gravitational forces or frictional forces.

In the context of our problem, a box is pushed up an inclined plane, and we know the amount of work done to move the box. The box starts and ends at rest, which implies that there's no change in kinetic energy (the energy associated with motion). Therefore, all of the work done on the box is used to overcome the gravitational pull on the box (which is trying to pull it back down the incline) as well as any frictional forces between the box and the plane. To calculate the applied force necessary to move the box, we use the equation:
\[W = F_{app}d \cdot \cos(\theta)\]
where \(W\) is the work done, \(F_{app}\) is the applied force, \(d\) is the distance the box moves (the length of the incline), and \(\theta\) is the angle of the incline relative to the horizontal.

Grasping this principle is key to solving problems related to forces and energy, and helps us understand how much energy is required to overcome resistance to motion.

The Role of Inclined Planes in Physics Problems

An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other. In physics, it's a classic example of a simple machine which can help lift or lower loads by applying forces more advantageously.

In our exercise, the box being pushed up an inclined plane requires less force to move vertically than if it was being lifted straight up because the force is spread out over a longer distance. The steeper the incline, the more force is required to move an object up the plane.

To find the angle of our plane, we use trigonometry:
\[\sin(\theta) = \frac{height}{length}\]
By finding the angle \(\theta\), we can determine the component of gravitational force acting parallel to the surface of the incline and thus understand how the angle affects the difficulty of the task.

Breaking Down Gravitational Force Components

Gravity pulls objects straight down towards the Earth's center. However, when dealing with an inclined surface, it is helpful to break down this gravitational force into two components: one that is perpendicular to the surface and one that is parallel to the surface.

The component of the gravitational force acting parallel to the inclined plane is responsible for the tendency of objects to slide down the plane. It is calculated as:
\[F_{g,\parallel} = mg\sin(\theta)\]
where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity (9.81 m/s²), and \(\theta\) is the angle of the inclined plane.

Understanding these components is crucial because it influences other forces at play, such as the normal force exerted by the plane on the object and the frictional force, which is our next topic.

Average Friction Force Calculation

Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. It plays a significant role when an object moves (or tries to move) across another surface. In our exercise, friction is the force that opposes the motion of the box as it is pushed up the incline.

To calculate the average frictional force, we consider the sum of forces parallel to the inclined plane. Since we know the box moves at a constant velocity (starts and ends at rest), the total force parallel to the inclined plane must be zero. Here, we have:
\[F_{friction} = F_{app,\parallel} - F_{g,\parallel}\]
We've already calculated the parallel component of the applied force and gravitational force. By substituting these values into our friction force equation, we can solve for the average friction force that acts on the box throughout its motion. This calcution enables us not only to understand the magnitude of the frictional force but also to explore concepts such as friction coefficients in more complex problems.