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Chapter 6: Problem 24

The gravitational potential energy of a person on a 3.0-m-high diving board is 1800 J. What is the person鈥檚 mass?

Short Answer

Expert verified
The person's mass is approximately 61.22 kg.

Step by step solution

01

Understand the Formula

The gravitational potential energy (GPE) is given by the formula \( GPE = m \cdot g \cdot h \) where \( m \) is the mass in kilograms, \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \) on Earth), and \( h \) is the height in meters. In this problem, we know the GPE and the height, and we want to find the mass.
02

Rearrange the Formula

We need to solve for the mass \( m \). Starting with the formula \( GPE = m \cdot g \cdot h \), we can rearrange it to \( m = \frac{GPE}{g \cdot h} \) by dividing both sides by \( g \cdot h \).
03

Substitute Known Values

Substitute the known values into the rearranged formula: \( GPE = 1800 \text{ J} \), \( g = 9.8 \text{ m/s}^2 \), and \( h = 3.0 \text{ m} \). Thus, \( m = \frac{1800}{9.8 \times 3.0} \).
04

Perform the Calculation

Calculate the mass using the substituted values: \( m = \frac{1800}{29.4} \). Divide 1800 by 29.4 to find \( m \approx 61.22 \text{ kg} \).

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Key Concepts

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

Mass Calculation
To determine the mass of a person based on their gravitational potential energy, we need to use a specific physics formula. Mass calculation is done by rearranging the formula for gravitational potential energy. In this context, the formula is given as:
  • \( GPE = m \cdot g \cdot h \)
  • where \( GPE \) is gravitational potential energy in joules, \( m \) represents mass in kilograms, \( g \) is the acceleration due to gravity, and \( h \) stands for height in meters.
Since our goal is to find the mass \( m \), we rearrange the formula to isolate \( m \):
  • \( m = \frac{GPE}{g \cdot h} \)
Rearranging StepsTo rearrange the equation, simply divide both sides of the original equation \( GPE = m \cdot g \cdot h \) by \( g \cdot h \). This isolates \( m \) allowing us to solve for mass when the values of gravitational potential energy, gravity, and height are known.
Physics Formulas
Physics often uses formulas to describe the relationships between different physical quantities. In the case of gravitational potential energy, it's crucial to understand what each part of the formula represents. In the gravitational potential energy formula:
  • \( GPE \) stands for gravitational potential energy, usually measured in joules (J).
  • \( m \) is the mass, in kilograms (kg), of the object or person.
  • \( g \) or the acceleration due to gravity, is about \( 9.8 \text{ m/s}^2 \) on Earth, which is a constant value.
  • \( h \) is the height, in meters (m), from which the mass is positioned.
Using the FormulaThis formula is essential for converting and understanding potential energy associated with an object's position relative to Earth. By having at least two known values from \( GPE, m, g, h \), we can determine the unknown variable.
Energy Conversion
Gravitational potential energy represents the energy stored due to an object's position in a gravitational field. This type of energy is directly related to mass and height. When dealing with gravitational potential energy, it's important to understand how energy can convert between different forms. How Energy Converts
  • For example, as a person stands on a diving board, they possess gravitational potential energy.
  • When they dive off, this energy converts to kinetic energy as they fall.
  • The principle of energy conservation states that energy in a closed system cannot be created or destroyed, just converted from one form to another.
This conversion helps explain why the initial gravitational potential energy affects the speed at which an object falls, emphasizing the interchangeable nature of various forms of energy.

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Most popular questions from this chapter

A block with a mass of \(3.7 \mathrm{~kg}\) slides with a speed of \(2.2 \mathrm{~m} / \mathrm{s}\) on a frictionless surface. The block runs into a stationary spring and compresses it a certain distance before coming to rest. What is the compression distance, given that the spring has a spring constant of \(3200 \mathrm{~N} / \mathrm{m}\) ?

Analyze You throw a baseball glove straight upward to celebrate a victory. Its initial kinetic energy is \(K E\), and it reaches a maximum height \(h\). What is the kinetic energy of the glove when it is at the height \(h / 2\) ?

Predict \& Explain When a ball of mass \(m\) is dropped from rest from a height \(h\), its kinetic energy just before landing is \(K E\). Now, suppose a second ball of mass \(4 m\) is dropped from rest from a height \(h / 4\). (a) Just before ball 2 lands, is its kinetic energy \(4 K E, 2 K E, K E, K E / 2\), or \(K E / 4\) ? (b) Choose the best explanation from among the following: A. The two balls have the same initial energy. B. The more massive ball will have the greater kinetic energy. C. The lower drop height results in a reduced kinetic energy.

A spring has a spring constant of \(310 \mathrm{~N} / \mathrm{m}\). Plot the potential energy for this spring when it is stretched by \(1.0 \mathrm{~cm}, 2.0 \mathrm{~cm}, 3.0 \mathrm{~cm}\), and \(4.0 \mathrm{~cm}\). Draw a curve that goes through your plotted points.

An object moves with no friction or air resistance. Initially, its kinetic energy is \(10 \mathrm{~J}\), and its gravitational potential energy is \(20 \mathrm{~J}\). What is its kinetic energy when its potential energy has decreased to \(15 \mathrm{~J}\) ? What is its potential energy when its kinetic energy has decreased to 5 J?
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