Question

In a 100 -person ski lift, a machine raises passengers averaging \(667 \mathrm{~N}\) in weight a height of \(152 \mathrm{~m}\) in \(55.0 \mathrm{~s}\), at constant speed. Find the power output of the motor, assuming no frictional losses.

Short answer

The power of the motor, when there are no frictional losses, is approximately 247.16 horsepower.

Step-by-step solution

Calculate the total force that is being lifted

In this step, the total force has to be calculated which is being lifted by the ski lift. This force is equal to the weight of all the people on the ski lift. To find the total force, first, calculate the force exerted by one person by multiplying the average weight (667N) with the total number of people on the ski lift. This results in the total force \( F = 667N/person \times 100 persons = 66700N\).

Compute the work done

The work done (W) lifting up the passengers can be calculated by multiplying the total force (F) and the vertical displacement or height (h). In other words, \( W = F \times h \). Using the values from step 1 and the given height value of 152m in the exercise, the work done is \( W = 66700N \times 152m = 10142400J \).

Find the power output

Power (P) is defined as the work done (W) per unit of time (t). Hence, we calculate the power by dividing the work done by the time it takes to lift the passengers. Hence, \( P = W/t \). Substituting the work done from Step 2 (10142400J) and the given time (55.0s) from the exercise, the power of the motor can be calculated as \( P = 10142400J / 55.0s = 184408.18W \).

Convert the power to more suitable units

Typically, the power of engines or motors is expressed in units of horsepower. 1 horsepower is approximately equal to 746 watts. Thus, to convert the calculated power from watts to horsepower we use the conversion factor \( 1hp = 746W \), resulting in the power output being \( P = 184408.18W / 746W/hp = 247.16hp \).

Key concepts

Work and Energy

Understanding the concepts of work and energy is fundamental in physics, particularly when analyzing mechanical systems. Work is defined as the transfer of energy to an object by applying a force that causes the object to move. It is mathematically expressed as the product of force and the displacement in the direction of the force, which is given by the equation:
\[ W = F \times d \] where W is work, F is the force exerted, and d is the displacement in meters.

Energy, on the other hand, is the ability to do work. It comes in various forms like kinetic energy, potential energy, thermal energy, etc. In the context of our ski lift problem, the energy is gravitational potential energy gained by the passengers as they are raised to a higher altitude. This energy change is equivalent to the work done by the ski lift's motor. With no frictional loss, all the work performed by the lift is transformed into potential energy of the passengers, helping us to calculate the required motor power output.

Practical Implications of Work

In real-world applications, understanding work is crucial for designing machines and structures. Knowing how much work is needed to raise a weight helps engineers select appropriate motors that can handle the load, just as we calculated for the ski lift.

Power Output Calculation

Power output calculation is an essential concept for determining how quickly work can be done, or equivalently, energy is transferred over time. It's often necessary in physics and engineering to know the rate at which a system can perform work, especially when designing machinery or electrical devices. The power output is expressed in watts (W) and calculated by the formula:
\[ P = \frac{W}{t} \] where P is power, W is work in joules, and t is time in seconds. To find the power output of our ski lift's motor, we simply divide the work done by the time taken to lift the passengers. The result tells us how efficient the lift is at converting electrical (or any other form of) energy into mechanical work.

Real-World Application

Knowing the power output is vital in sectors like automotive, where engines are rated based on how quickly they can convert fuel into kinetic energy to move a vehicle. In our ski lift example, the higher the power output, the quicker the lift can operate, enhancing the overall efficiency and capacity of the lift system.

Forces in Physics

Forces are central to the study of physics, encapsulating one of the basic interactions in the universe. In the case of our ski lift, the force of interest is weight, a force due to gravity acting on a mass. The weight of an object is calculated as the product of mass and the acceleration due to gravity, which is approximately \(9.81 m/s^2\) on Earth. In the exercise, the weight of a person is already given as a force in newtons (N), thus streamlining our calculation as we do not need to consider individual masses.

A proper understanding of forces can help us analyze and predict the motion of objects and systems. For instance, frictional force — although not considered in this idealized ski lift scenario — often plays a significant role in real-world applications, opposing the motion and requiring additional power to overcome.

Essential Nature of Forces

Recall that forces can be a combination of more than one force, known as the net force. This approach allows us to understand complex systems by simplifying the multitude of forces into a single, resultant force that can then be used in calculations of work and energy.