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Chapter 6: Problem 3

You want to raise a piano a given height using a ramp. With a fixed, nonzero coefficient of friction, will you have to do more work if the ramp is steeper or more gradual? Explain.

Short Answer

Expert verified
The work done to move the piano up the ramp remains the same whether the ramp is steeper or more gradual. This is because the increase in work done against friction with a steeper ramp is offset by the shorter distance, and vice versa.

Step by step solution

01

Understanding the Concept of Work

Work is defined as the force exerted on an object times the distance moved by the object, generalized as \( W = Fd \). When raising a piano up a ramp with friction, the work will be equal to the work against gravity plus the work against friction.
02

Consider the Work Against Gravity

The piano's work against gravity remains constant regardless of the slope of the ramp. This is because the change in the piano鈥檚 gravitational potential energy depends on the height it is raised, and not on the path taken. The work done against gravity doesn't change.
03

Consider the Work Against Friction

The friction does depend on the path taken because friction is a non-conservative force. The work against friction is calculated by the formula \( F_f * d \), wherein \( F_f \) refers to the force of friction and \( d \) is the distance over which the force is applied. With a longer (more gradual) ramp, the normal force, and therefore friction, is less, but the distance is greater. In contrast, with a steeper ramp, the normal force, and therefore the friction, is more, but the distance is less.
04

Establishing the Relationship Between the Steepness of the Ramp and Work Done

The overall work done in moving the piano is a sum of the work done against gravity and friction. Given that the mutual magnitude of friction and the distance makes their product (which represents work done against friction) variables according to the steepness, the total work done doesn't remain constant. However, the change in potential energy (work done against gravity) remains the same. Therefore, regardless of whether the ramp is steeper or more gradual, the overall work done remains the same.

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