Question

A small motor runs a lift that raises a load of bricks weighing \(836 \mathrm{~N}\) to a height of \(10.7 \mathrm{~m}\) in \(23.2 \mathrm{~s}\). Assuming that the bricks are lifted with constant speed, what is the minimum power the motor must produce?

Short answer

The motor must produce at least 385.6 W of power.

Step-by-step solution

Understand the Problem

We need to find the minimum power required by the motor to lift a load weighing \(836\, \text{N}\) to a height of \(10.7\, \text{m}\) in \(23.2\, \text{s}\). The power refers to the rate at which work is done.

Calculate the Work Done

The work done in lifting the load can be calculated using the formula: Work = Force \(\times\) Distance. Here, the force is the weight of the bricks and the distance is the height they are lifted. Thus, Work = \(836\, \text{N} \times 10.7\, \text{m} = 8945.2\, \text{J}\).

Determine the Time in Seconds

We know from the problem statement that the time taken to lift the bricks is \(23.2\, \text{s}\). This time will be used to calculate the power.

Calculate the Power

Power is defined as the rate of doing work. It is given by the formula: Power = Work / Time. Using our calculated work and the time, we have Power = \(8945.2\, \text{J} / 23.2\, \text{s} = 385.6\, \text{W}\).

State the Final Answer

The minimum power the motor must produce to lift the bricks at a constant speed is \(385.6\, \text{W}\).

Key concepts

Understanding Work Done

When we talk about 'work done' in physics, we are referring to the amount of energy transferred when a force is applied to move an object over a distance. For this exercise, the motor did work by moving bricks upward. The formula for calculating work is:

• Work = Force \( \times \) Distance

The force in this scenario is the weight of the bricks, which happens to be the force due to gravity acting on them. Hence, it's measured in newtons (N). The distance is how high the bricks were lifted, measured in meters (m). Applying the numbers from the exercise, we have:

• Force = 836 N
• Distance = 10.7 m
• Work Done = 836 N \( \times \) 10.7 m = 8945.2 J (joules)

Essentially, this tells us the total energy used in lifting the bricks.

The Relationship Between Force and Motion

Force and motion have a fundamental connection in physics, as described by Newton's laws. In this exercise, the force is the push or pull exerted by the motor to lift the bricks against gravity. Since the problem states that bricks are lifted at a constant speed, this means:

• The net force acting on the bricks is zero.
• The force exerted by the motor is equal to the gravitational force acting downward.

This balance of forces implies that the motor supplies only as much force as needed to counter gravity. Thus, if you increase the force, you'd accelerate the load, while too little force would cause it to fall. Understanding the balance of force and motion helps us calculate the exactly required force to maintain that constant upward motion without extra acceleration.

Exploring the Rate of Work or Power

Power is a measure of how quickly work is done or energy is transferred. The faster the work is performed, the higher the power. In this exercise, we were required to determine the minimum power the motor must produce to lift bricks. The formula for power is:

• Power = Work / Time

The work done on the bricks is already calculated, and the time taken was given as 23.2 seconds:

• Work Done = 8945.2 J
• Time = 23.2 s
• Power = 8945.2 J / 23.2 s ≈ 385.6 W (watts)

This calculation shows the rate at which energy is used to lift the bricks. A higher wattage would mean faster lifting capability or a heavier load capacity. Power is crucial not only for understanding the capability of motors but also for any mechanical work involving energy transfer over time.