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Chapter 5: Problem 3

Furniture movers wish to load a truck using a ramp from the ground to the rear of the truck. One of the movers claims that less work would be required if the ramp's length were increased, reducing its angle with the horizontal. Is this claim valid? Explain.

Short Answer

Expert verified
The mover's claim that less work would be required if the ramp's length were increased, reducing its angle with the horizontal, is not valid in terms of physics, as the work done against gravity remains the same. However, the force required to move the furniture could be perceived as 'less work' because it is less over a longer distance.

Step by step solution

01

Calculate the work done with the original ramp

The work done to move the furniture up the ramp can be calculated by multiplying the weight of the furniture \(W=mg\) by the vertical distance \(h\) it has been lifted, i.e., \(Work_1 = Wh\).
02

Calculate the work done with the ramp at reduced angle

When the ramp's angle is reduced, it becomes longer. This means that the same piece of furniture must be moved over a greater distance. However, because the ramp's height \(h'\) has not changed, only its length, the work done against gravity remains the same. Hence, \(Work_2 = mgh'\), but since \(h=h'\), therefore, \(Work_2 = Wh = Work_1\). The total work done is the same, but the mover might need to exert less force over a longer distance.
03

Compare the work done in both scenarios

Since the work done in both scenarios is the same, the mover's claim that less work would be required if the ramp's length were increased, reducing its angle with the horizontal, is not valid. However, the force required to move the furniture might be less, because the force required to move an object up a slope is inversely proportional to the length of the slope. Therefore, a longer ramp would mean a smaller force is required to move the furniture, which could be perceived as 'less work' by the mover.

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Key Concepts

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

Inclined Plane
An inclined plane is one of the basic types of simple machines that make moving heavy loads a bit easier. It is essentially a flat surface tilted at an angle to the horizontal. Instead of lifting an object straight up, which requires a lot of force, the inclined plane allows you to push it along the slanted surface, effectively spreading out the force over a greater distance. This is why ramps are often used for moving heavy items like furniture into trucks or buildings.
  • The angle of the inclined plane can vary, which affects how much force is needed to push the object up the slope.
  • Changing the length of the plane changes the angle. A longer ramp means a smaller angle, hence reducing the steepness.
The important takeaway here is that an inclined plane redistributes the force needed, even if it doesn't change the amount of work needed.
Mechanical Advantage
Mechanical advantage refers to the factor by which a machine multiplies the force you exert. With an inclined plane, the mechanical advantage is gained by a longer slope. While the amount of work done remains constant (since it depends on the change in height and weight of the object), the force needed to move the object can be reduced.

When you use a longer ramp:
  • The mechanical advantage increases because the force you need to apply decreases.
  • It requires less force over a greater distance. As a result, the task feels easier even though the work done is the same.
This is crucial for understanding how ramps help in everyday tasks; although the work done doesn't change, the perception and ease of effort do, thanks to mechanical advantage.
Force and Distance Relationship
The relationship between force and distance is a fundamental concept in physics. When dealing with inclined planes, this relationship explains why a longer ramp requires less force to move the same object to the same height. This is based on the work-energy principle, which states that work is the product of force and distance.
  • Work (W) is calculated by multiplying the force (F) applied by the distance (d) over which the force is applied: \[ W = F \times d \]
  • Even if the distance on the ramp is increased (by making the ramp longer), the total work required to raise the object remains constant.
By extending the distance over which force is applied (a longer ramp), the force needed at any given moment is reduced. This trade-off between force and distance is why movers perceive that they are doing less work, when in fact, the work done remains unchanged.

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Most popular questions from this chapter

A light horizontal spring has a spring constant of \(105 \mathrm{N} / \mathrm{m} .\) A \(2.00 \mathrm{kg}\) block is pressed against one end of the spring, compressing the spring \(0.100 \mathrm{m}\). After the block is released, the block moves \(0.250 \mathrm{m}\) to the right before coming to rest. What is the coefficient of kinetic friction between the horizontal surface and the block?

A person doing a chin-up weighs \(700.0 \mathrm{N},\) disregarding the weight of the arms. During the first \(25.0 \mathrm{cm}\) of the lift, each arm exerts an upward force of \(355 \mathrm{N}\) on the torso. If the upward movement starts from rest, what is the person's speed at this point?

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Starting from rest, a \(5.0 \mathrm{kg}\) block slides \(2.5 \mathrm{m}\) down a rough \(30.0^{\circ}\) incline in 2.0 s. Determine the following: a. the work done by the force of gravity b. the mechanical energy lost due to friction c. the work done by the normal force between the block and the incline

Three identical balls, all with the same initial speed, are thrown by a juggling clown on a tightrope. The first ball is thrown horizontally, the second is thrown at some angle above the horizontal, and the third is thrown at some angle below the horizontal. Disregarding air resistance, describe the motions of the three balls, and compare the speeds of the balls as they reach the ground.
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