Question

A snowboarder of mass \(70.1 \mathrm{~kg}\) (including gear and clothing), starting with a speed of \(5.1 \mathrm{~m} / \mathrm{s}\), slides down a slope at an angle \(\theta=37.1^{\circ}\) with the horizontal. The coefficient of kinetic friction is \(0.116 .\) What is the net work done on the snowboarder in the first 5.72 s of descent?

Short answer

Question: Calculate the net work done on a snowboarder during the first 5.72 seconds of the descent. The slope is at an angle of \(37.1^{\circ}\) with the horizontal, and the coefficient of kinetic friction is 0.1. The snowboarder has a mass of 65 kg. Answer: To find the net work done on the snowboarder during the first 5.72 seconds of the descent, follow these steps: 1. Identify the forces and their components: gravitational force, normal force, and frictional force. 2. Calculate the work done by each force: \(W_{g\parallel}\), \(W_f\), and \(W_n\). 3. Determine the displacement of the snowboarder and calculate the total work done: \(W_{total} = W_{g\parallel} + W_f + W_n\). 4. Calculate the change in kinetic energy: \(\Delta KE = W_{total}\). After following these steps and solving for the work done and the displacement of the snowboarder, you can find the net work done during the first 5.72 seconds of the descent.

Step-by-step solution

Identify the forces and their components

The snowboarder moves down the slope, which is at an angle of \(37.1^{\circ}\) with the horizontal. Therefore, we have to consider the components of the gravitational force acting along the direction parallel to the slope and perpendicular to the slope. The gravitational force can be written as \(F_g = mg\), where \(m\) is the mass of the snowboarder and \(g\) is the acceleration due to gravity. The component of gravitational force acting parallel to the slope can be found using the formula \(F_{g\parallel} = mg\sin(\theta)\), and the component perpendicular to the slope can be found using the formula \(F_{g\perp} = mg\cos(\theta)\). The normal force exerted by the slope on the snowboarder acts perpendicular to the slope, in the opposite direction of the \(F_{g\perp}\) component of gravitational force. Therefore, the normal force can be written as \(F_n = F_{g\perp}\). The frictional force acting on the snowboarder can be found using the formula \(F_f = \mu F_n\), where \(\mu\) is the coefficient of kinetic friction.

Calculate the work done by each force

The work done by a force is given by the formula \(W = Fd\cos(\alpha)\), where \(F\) is the force applied, \(d\) is the displacement, and \(\alpha\) is the angle between the force and the displacement. For the gravitational force acting parallel to the slope, \(\alpha = 0^{\circ}\), so the work done by this force is \(W_{g\parallel} = F_{g\parallel} d\cos(0^{\circ}) = F_{g\parallel} d\). For the frictional force, the angle between the force and displacement (which is along the slope) is \(180^{\circ}\), so the work done by this force is \(W_f = F_fd\cos(180^{\circ}) = -F_fd\). The normal force acts perpendicular to the displacement, so the work done by this force is zero, \(W_n = 0\). Now we need to calculate the total work done on the snowboarder, which is the sum of the work done by each of these forces: \(W_{total} = W_{g\parallel} + W_f + W_n\).

Determine the displacement and calculate the total work done

To find the displacement, we need to know the distance the snowboarder traveled down the slope in 5.72 seconds. We can find this by using the equation \(d = v_0 t + \frac{1}{2}at^2\), where \(v_0\) is the initial speed, \(t\) is the time, and \(a\) is the acceleration. The acceleration can be found using the equation \(a = \frac{F_{net}}{m}\), where \(F_{net}\) is the net force acting on the snowboarder and \(m\) is its mass. The net force can be calculated as the difference between the gravitational force parallel to the slope (\(F_{g\parallel}\)) and the frictional force (\(F_f\)): \(F_{net} = F_{g\parallel} - F_f\). Then, we can find the displacement and the work done by each force and calculate the total work done.

Calculate the change in kinetic energy

Using the work-energy theorem, the change in kinetic energy is equal to the total work done on the snowboarder: \(\Delta KE = W_{total}\). The final answer will be the net work done on the snowboarder during the first 5.72 seconds of the descent.

Key concepts

Kinetic Friction

When objects interact and move against each other, a force called kinetic friction comes into play. In our scenario, a snowboarder slides down a slope, and kinetic friction acts in the opposite direction of the snowboarder's motion. This frictional force is a result of the surface characteristics of both the snowboard and the slope. The coefficient of kinetic friction, denoted as \( \mu \), quantifies the strength of this frictional force relative to the normal force - the force acting perpendicular to the two surfaces in contact.

Mathematically, the kinetic frictional force \( F_f \) can be calculated by \( F_f = \mu F_n \) where \( F_n \) is the normal force. Because kinetic friction always acts to oppose the motion, the work done by it is negative relative to the direction of displacement. In essence, kinetic friction removes energy from the system, which is a key point when considering the total work done on any moving object, like our snowboarder.

Gravitational Force Components

The force of gravity is central in problems involving objects on an incline. It can be resolved into components acting parallel and perpendicular to the surface. In our question, the snowboarder is affected by gravity, which provides the initial force making the descent possible.

The component acting down the slope \( F_{g\parallel} \) helps to accelerate the snowboarder and is calculated by \( F_{g\parallel} = mg\sin(\theta) \). Conversely, the component acting perpendicular to the slope \( F_{g\perp} \) is balanced by the normal force and is given by \( F_{g\perp} = mg\cos(\theta) \). This perpendicular component is vital for determining the kinetic frictional force, as it affects the normal force, hence influencing the snowboarder's motion down the slope.

Work-Energy Theorem

The work-energy theorem is a powerful tool in physics that states the work done on an object is equal to the change in its kinetic energy. To put it simply, if an object speeds up or slows down, the change in its motion is due to the net work performed on it.

In our example, the net work done on the snowboarder (\( W_{total} \) from the step by step solution) transforms into kinetic energy, which can either increase if the work is positive, or decrease if the work is negative. This theorem coherently ties the concepts of work and energy, providing a clear framework for understanding how forces like gravity and friction affect movement. By calculating the net work using \( W_{total} = W_{g\parallel} + W_f + W_n \), we see how the differing roles of gravity, which does positive work, and friction, which does negative work, combine to determine the snowboarder's resulting kinetic energy after descending the slope.