Question

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

The gravitational potential energy is approximately 121.474 J.

Step-by-step solution

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 \(9.8 \text{ m/s}^2\).

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.

Substitution into the Formula

Substitute the given values into the formula: \[U = (7.3 \text{ kg}) \times (9.8 \text{ m/s}^2) \times (1.7 \text{ m})\]

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 \(121.474 \text{ joules}\).

Key concepts

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.

When calculating the gravitational potential energy, the mass directly influences the total energy: a heavier object has more gravitational potential energy than a lighter one, provided they are at the same height. Understanding mass is essential in physics because it also relates to an object's inertia, or its resistance to changes in motion. However, remember that when evaluating potential energy at heights, it is the object's weight (derived from mass and gravity) that matters most.

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.

Remember that this value is a simplification, as gravity can slightly vary at different altitudes and locations on Earth.

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.

For any calculations involving potential energy, always ensure the height is accurately measured from the specified reference point to avoid errors. The clearer you are about the starting and reference points, the more precise your calculations will be.

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)

To calculate, simply multiply these three values:1. Multiply the mass by the acceleration: 7.3 kg × 9.8 m/s²2. Take that result and multiply by the height: (7.3 kg × 9.8 m/s²) × 1.7 mThis process results in the gravitational potential energy:\[U = 121.474 \]This calculation shows an energy of about **121.474 joules**, indicating the stored energy due to the object's elevated position. Such a straightforward yet powerful formula helps to connect different physical concepts and form a clearer understanding of energy transformations.