Chapter 6: Problem 86
A 7.3-kg bowling ball is placed on a shelf \(1.7 \mathrm{~m}\) above the floor. What is its gravitational potential energy?
Short Answer
Expert verified
The gravitational potential energy is approximately 121.474 J.
Step by step solution
01
Identify the Given Values
According to the problem, we have a bowling ball with a mass \(m = 7.3 \text{ kg}\) and it is raised to a height \(h = 1.7 \text{ m}\) above the floor. The acceleration due to gravity \(g\) is approximately \(9.8 \text{ m/s}^2\).
02
Gravitational Potential Energy Formula
The formula to calculate the gravitational potential energy \(U\) is \[U = mgh\]where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity, and \(h\) is the height above the reference point.
03
Substitution into the Formula
Substitute the given values into the formula: \[U = (7.3 \text{ kg}) \times (9.8 \text{ m/s}^2) \times (1.7 \text{ m})\]
04
Calculate the Gravitational Potential Energy
Multiply the values to find \(U\):\[U = 7.3 \times 9.8 \times 1.7 = 121.474 \text{ J}\]Thus, the gravitational potential energy of the bowling ball is approximately \(121.474 \text{ joules}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mass
Mass refers to the amount of matter in an object. In the context of gravitational potential energy, it is a crucial factor in determining how much energy an object possesses when it is elevated to a certain height. The mass of an object is usually measured in kilograms (kg). For example:
- A feather has a small mass.
- A bowling ball, as in the given exercise, has a larger mass of 7.3 kg.
Acceleration due to Gravity
Gravity is a force that attracts objects towards each other. On Earth, this force pulls everything towards the center of the planet. The acceleration due to gravity (
), denoted as **g**, is approximately **9.8 meters per second squared (m/s²)** here. This constant is vital in many physics calculations, particularly for gravitational potential energy.
Since the force of gravity is what gives weight to an object, acceleration due to gravity acts as a bridge in calculations between mass and force. Formulaically, it helps determine how quickly an object will accelerate downward if dropped. For potential energy:
- It combines with mass to define weight, influencing how much potential energy an object has when lifted.
- In our exercise, it's used to calculate the energy stored when the bowling ball is elevated above the floor.
Height
Height (
h
) refers to the vertical distance an object is lifted above a reference point, like the ground or a floor. It plays a key role in calculating gravitational potential energy, because potential energy increases with greater heights. Height is measured in meters (m), and in our exercise, it's 1.7 m above the floor.
Here's how height affects potential energy:
- An object raised higher will have more potential energy as there's more distance for gravity to "act" on it if dropped.
- Even with the same mass, increasing the height raises the energy potential.
Energy Calculation
Energy calculation in this context involves determining the gravitational potential energy (U) of an object. The formula used is:\[U = mgh\]where:
- **m** = mass of the object (7.3 kg for our bowling ball)
- **g** = acceleration due to gravity (9.8 m/s² on Earth)
- **h** = height above the reference point (1.7 m above the floor)