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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

Expert verified
The motor must produce at least 385.6 W of power.

Step by step solution

01

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.
02

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 = \(836\, \text{N} \times 10.7\, \text{m} = 8945.2\, \text{J}\).
03

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.
04

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 = \(8945.2\, \text{J} / 23.2\, \text{s} = 385.6\, \text{W}\).
05

State the Final Answer

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A spring that is stretched \(2.6 \mathrm{~cm}\) stores a potential energy of \(0.053 \mathrm{~J}\). What is the spring constant of this spring?

A \(51-\mathrm{kg}\) packing crate is pulled across a rough floor with a rope that is at an angle of \(43^{\circ}\) above the horizontal. If the tension in the rope is \(120 \mathrm{~N}\), how much work is done on the crate to move it \(18 \mathrm{~m}\) ?

A player passes a \(0.600-\mathrm{kg}\) basketball down court for a fast break. The ball leaves the player's hands with a speed of \(8.30 \mathrm{~m} / \mathrm{s}\) and slows down to \(7.10 \mathrm{~m} / \mathrm{s}\) at its highest point. Ignoring air resistance, how high above the release point is the ball when it is at its maximum height?

Think \& Calculate A sled slides without friction down a small, ice-covered hill. If the sled starts from rest at the top of the hill, its speed at the bottom is \(7.50 \mathrm{~m} / \mathrm{s}\). (a) On a second run, the sled starts with a speed of \(1.50 \mathrm{~m} / \mathrm{s}\) at the top. When it reaches the bottom of the hill, is its speed \(9.00 \mathrm{~m} / \mathrm{s}\), more than \(9.00 \mathrm{~m} / \mathrm{s}\), or less than \(9.00 \mathrm{~m} / \mathrm{s}\) ? Explain. (b) Find the speed of the sled at the bottom of the hill after the second run.

Calculate The coefficient of kinetic friction between a large box and the floor is 0.21. A person pushes horizontally on the box with a force of 160 N for a distance of 2.3 m. If the mass of the box is 72 kg, what is the total work done on the box?

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