Chapter 6: Problem 86
A \(7.3-\mathrm{kg}\) 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 121.6 Joules.
Step by step solution
01
Understand the Formula
The gravitational potential energy (U) of an object is given by the formula:\[ U = m \cdot g \cdot h \]where:- \( m \) is the mass of the object in kilograms,- \( g \) is the acceleration due to gravity (approximately \( 9.8 \ \mathrm{m/s^2} \) on Earth), and- \( h \) is the height above the reference point in meters.
02
Identify the Given Values
From the problem, we have:- Mass ( m ) = 7.3 kg,- Height ( h ) = 1.7 m.- Use \( g \) as 9.8 \( \mathrm{m/s^2} \) for the acceleration due to gravity.
03
Plug in the Values
Substitute the given values into the gravitational potential energy formula:\[ U = 7.3 \ \mathrm{kg} \times 9.8 \ \mathrm{m/s^2} \times 1.7 \ \mathrm{m} \]
04
Perform the Calculation
Calculate the potential energy:First, calculate the product of the mass and gravity:\[ 7.3 \ \mathrm{kg} \times 9.8 \ \mathrm{m/s^2} = 71.54 \ \text{Nm (Newton meters)} \]Then, multiply by the height:\[ 71.54 \ \text{Nm} \times 1.7 \ \mathrm{m} = 121.618 \ \text{Nm} \]
05
State the Result
Round the result to a reasonable number of significant figures. The gravitational potential energy is approximately:\[ U \approx 121.6 \ \text{Joules} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Physics Problem Solving
Physics problem solving involves breaking down complex ideas into simpler components and applying the right formulas to find a solution. Let's consider the process to solve for gravitational potential energy in the problem:
- Identify the relevant physical quantities in the question. Here, these include mass, acceleration due to gravity, and height.
- Select the appropriate formula. In this case, the potential energy formula is used.
- Substitute the given values into the formula and calculate. Ensure units are consistent.
- Check your final answer for reasonableness, including significant figures, as shown in the solution where we rounded the potential energy to 121.6 Joules.
Kinematics
While gravitational potential energy isn't directly about motion, it's deeply tied to kinematics — the study of motion. Potential energy is related to the position of an object relative to a gravitational field. In this problem:
- Consider the height as a key kinematic variable.
- Kinematics also involves motion influenced by forces. Here, gravity is a force impacting the potential energy when the ball is elevated.
Newton's Laws
Newton's laws of motion connect closely with the concept of gravitational potential energy. Particularly relevant is the law of universal gravitation:
- According to Newton's second law, force equals mass times acceleration. Here, the force is gravitational, with acceleration due to gravity.
- The gravitational force acts on the mass of the bowling ball, leading to potential energy as it's raised against gravity's pull.
- This exercise also reflects Newton's third law. As the ball has gravitational potential energy, when it is released, gravitational force will cause it to fall, demonstrating action and reaction.