Question

A horse draws a sled horizontally across a snowcovered field. The coefficient of friction between the sled and the snow is \(0.195,\) and the mass of the sled, including the load, is \(202.3 \mathrm{~kg}\). If the horse moves the sled at a constant speed of \(1.785 \mathrm{~m} / \mathrm{s}\), what is the power needed to accomplish this?

Short answer

Based on the given step by step solution, we can create the following short answer: To calculate the power needed to move the sled at a constant speed, we must first determine the force of friction, which is found by multiplying the friction coefficient by the normal force. The normal force can be found by multiplying the mass of the sled by the gravitational constant. Next, we find the work done by multiplying the friction force by the distance moved, which can be represented by the product of the constant speed and the time. Then, we derive the power needed by dividing the work done by the time, which simplifies to the product of the friction force and the constant speed. Finally, we plug in the values to find the required power.

Step-by-step solution

Find the force of friction

To find the force of friction acting on the sled, we use the formula: Friction force (F_f) = friction coefficient (μ) × normal force (F_N) Since the sled is moving horizontally, the normal force is equal to its weight, which can be calculated as: Normal force (F_N) = mass (m) × gravity (g) F_N = 202.3 kg × 9.81 m/s²

Calculate the friction force

Now that we know the normal force, we can calculate the friction force using the given coefficient of friction: F_f = μ × F_N F_f = 0.195 × (202.3 kg × 9.81 m/s²)

Find the work done

To move the sled at a constant speed, the horse must overcome the friction force, so the work done is equal to the product of the force and the distance the sled moves, which we can represent as D. The distance can be found by multiplying the constant speed (v) by the time (t) the sled is in motion: D = v × t In this case, we're looking for the power (P) needed to accomplish this, so instead of finding the actual work done, we can express the work in terms of power and time: Work (W) = P × t

Calculate the power

Now that we know the work is equal to the product of power and time, we can find the power needed to move the sled. We'll divide both sides of the equation by t to get the power expression alone: P = W/t Since we know Work (W) = F_f × D, we can replace W with the expression: P = (F_f × D)/t We also know that distance (D) = v × t, so we can replace D in the expression: P = (F_f × v × t)/t Finally, we see that t cancels out, and we are left with: P = F_f × v Now we can plug in the values we found for friction force (F_f) and constant speed (v) to find the power needed: P = (0.195 × (202.3 kg × 9.81 m/s²)) × 1.785 m/s.

Key concepts

Understanding Friction Force

The concept of friction force is pivotal in various fields of physics and everyday life. It is the resistance that one surface or object encounters when moving over another. The force of friction always acts in the opposite direction of the movement.

There are several types of friction, but the one we're dealing with here is kinetic friction—the force that needs to be overcome to keep an object moving at a constant speed. The formula to calculate the kinetic friction force (\( F_f \)) is by multiplying the coefficient of friction (\( \text{μ} \)) by the normal force (\( F_N \)). The normal force is usually the weight of the object when movement is horizontal, which is the mass (\( m \)) of the object times the acceleration due to gravity (\( g \)), therefore: \( F_N = m \times g \).

These principles are not just academic; they are essential for engineers designing vehicles and machinery, for athletes understanding the performance of sports equipment, and even for calculating the energy needed for tasks, which leads us to the next concept—power calculation.

Power Calculation in Physics

The second key concept involved in this problem is power calculation. Power in physics is defined as the rate at which work is done or energy is transferred. It's how quickly or slowly a force can move an object over a certain distance. The unit of power is the watt (\( W \)), which is equal to one joule per second (\( J/s \)).

The formula to calculate power (\( P \)) when a constant force is applied over a distance at a constant speed is \( P = F \times v \), where \( F \) is the force applied and \( v \) is the velocity. In terms of our sled problem, the horse applies a force to overcome the frictional force and move the sled at a constant speed. The power needed for this action is the force of friction multiplied by the sled's velocity.

Understanding power calculation is crucial for a wide array of applications from designing motors and engines to determining the physical capabilities of humans and animals.

The Role of Constant Speed

In our sled-scenario, the horse maintains a constant speed. But what does it mean to travel at a constant speed? In physics terms, an object moves at a constant speed if it covers equal distances in equal intervals of time, regardless of the direction of travel.

This concept is important in the context of our problem, because if the speed is constant, the power output required is steady, not varying. Thus, the horse doesn't need to exert more force as time goes on to maintain the speed, despite friction acting against it. The concept of constant speed simplifies the calculation of work and power because if the velocity or speed changes, we would then need to consider the object's acceleration, complicating the power calculation.

Understanding constant speed helps not only in solving physics problems but also in real-life scenarios, like maintaining a steady pace when driving a car to increase fuel efficiency or in pacing during endurance sports.