Question

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

The gravitational potential energy is approximately 121.6 Joules.

Step-by-step solution

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.

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.

Plug in the Values

Substitute the given values into the gravitational potential energy formula:\[ U = 7.3 \ \mathrm{kg} \times 9.8 \ \mathrm{m/s^2} \times 1.7 \ \mathrm{m} \]

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:\[ 71.54 \ \text{Nm} \times 1.7 \ \mathrm{m} = 121.618 \ \text{Nm} \]

State the Result

Round the result to a reasonable number of significant figures. The gravitational potential energy is approximately:\[ U \approx 121.6 \ \text{Joules} \]

Key concepts

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.

The ability to solve physics problems efficiently grows with practice. By following systematic steps, such as those outlined above, more complex problems can be tackled with confidence.

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.

Understanding how height and potential energy relate helps comprehend energy conversion. When the bowling ball is released, potential energy transforms into kinetic energy, characterized by motion and velocity. These relations illustrate why mastering kinematics is crucial, as it allows for deeper understanding of energy interactions in various contexts.

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.

In essence, Newton's laws provide the framework for understanding why and how energy like gravitational potential manifests and affects objects. Recognizing these principles supports a solid foundation for tackling a range of physics problems efficiently.