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Chapter 6: Problem 32

a) If the gravitational potential energy of a 40.0 -kg rock is \(500 .\) J relative to a value of zero on the ground, how high is the rock above the ground? b) If the rock were lifted to twice its original height, how would the value of its gravitational potential energy change?

Short Answer

Expert verified
Answer: The height of the rock above the ground is approximately 1.27 meters. When the rock is lifted to twice its original height, the gravitational potential energy doubles to 1000 J.

Step by step solution

01

Write down the known values and the formula for GPE

We are given: Mass of rock (m) = 40.0 kg Gravitational potential energy (GPE) = 500 J Acceleration due to gravity (g) ≈ 9.81 m/s² (we will use this value) The formula for GPE: \(GPE = m \times g \times h\)
02

Solve for height (h) in part a

Start by rearranging the GPE formula to solve for h: \(h = \frac{GPE}{m \times g}\) Substitute the known values into the formula: \(h = \frac{500J}{40.0kg \times 9.81m/s^2}\) Now, calculate the height: \(h ≈ \frac{500}{40.0 \times 9.81} ≈ 1.27\)m The rock is 1.27 meters above the ground.
03

Calculate the new GPE when the rock is lifted to twice its original height in part b

First, find the new height by doubling the original height: \(new\_height = 2 \times 1.27m = 2.54m\) Now, use the GPE formula with the new height: \(new\_GPE = m \times g \times new\_height\) Substitute the known values into the formula: \(new\_GPE = 40.0kg \times 9.81m/s^2 \times 2.54m\) Now, calculate the new GPE: \(new\_GPE ≈ 40.0 \times 9.81 \times 2.54 ≈ 1000J\)
04

Compare the initial and new GPE

To determine how the GPE has changed when the rock was raised to twice its original height, we will now compare the initial GPE (500 J) and the new GPE (1000 J). The new GPE is twice the value of the initial GPE, which means that the gravitational potential energy has doubled when the rock was lifted to twice its original height.

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Key Concepts

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

GPE formula
Gravitational Potential Energy (GPE) is a type of potential energy that an object possesses due to its position in a gravitational field. The higher an object is above a reference point, like the ground, the greater its gravitational potential energy. The core of understanding this concept lies in the GPE formula, which directly relates mass, gravitational acceleration, and height to calculate an object's potential energy.

The standard formula for gravitational potential energy is expressed as: \[\begin{equation} GPE = m \times g \times h d{equation}
In this formula,
  • is the mass of the object in kilograms (kg), reminds us that heavier objects have greater GPE,
  • \(g\) is the acceleration due to gravity, which is approximately \(9.81 m/s^2\) on Earth's surface, and

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