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A \(2.0\) -kg mass falls \(400 \mathrm{~cm}\). (a) How much work was done on it by the gravitational force? ( \(b\) ) How much \(\mathrm{PE}_{\mathrm{G}}\) did it lose? (c) Given that work is the transfer of energy, where does that energy end up? (a) Gravity pulls with a force \(m g\) on the object, and the displacement is \(4 \mathrm{~m}\) in the direction of the force. The positive work done by gravity is therefore $$ (m g)(4.00 \mathrm{~m})=(2.0 \mathrm{~kg} \times 9.81 \mathrm{~N})(4.00 \mathrm{~m})=78 \mathrm{~J} $$ (b) The change in \(\mathrm{PE}_{\mathrm{G}}\) of the object is \(m g h_{f}-m g h_{i}\), where \(h_{i}\) and \(h_{f}\) are the initial and final heights of the object above the reference level. We then have $$ \begin{array}{l} \text { Change in } \mathrm{PE}_{\mathrm{G}}=m g h_{f}-m g h_{i}=m g\left(h_{f}-h_{i}\right)-(2.0 \mathrm{~kg} \times 9.81 \\ \mathrm{N})(-4.0 \mathrm{~m})=-78 \mathrm{~J} \end{array} $$ The loss in \(\mathrm{PE}_{\mathrm{G}}\) is \(78 \mathrm{~J}\). (c) Gravity provides the force that accelerates the \(2.0\) -kg mass and increases its kinetic energy by \(78 \mathrm{~J}\).

Step by step solution

01

Understanding Work Done by Gravity

To calculate the work done by gravity, we use the formula \( W = F \cdot d \cdot \cos(\theta) \), where \( F \) is the force, \( d \) is the distance, and \( \theta \) is the angle between the force and the direction of motion. Here, the force is gravity \( (mg) \) and the angle is 0 degrees as the force is in the same direction as the displacement. Hence, \( W = mg \cdot d \).

02

Calculating the Force of Gravity

The force of gravity \( F \) acting on the mass is given by \( F = mg \), where \( m = 2.0 \text{ kg} \) and \( g = 9.81 \text{ m/s}^2 \). This results in \( F = 2.0 \times 9.81 = 19.62 \text{ N} \).

03

Calculating the Work Done

Using the formula for work, \( W = mg \cdot d \), where \( d = 4 \text{ m} \), we calculate \( W = 19.62 \times 4.0 \). This gives \( W = 78 \text{ J} \).

04

Calculating Change in Potential Energy

The change in gravitational potential energy, \( \Delta \text{PE}_G \), is given by \( \Delta \text{PE}_G = mg(h_f - h_i) \), where \( h_i \) is the initial height and \( h_f \) is the final height. Since it falls 4 meters, \( h_f - h_i = -4 \). So, \( \Delta \text{PE}_G = 19.62 \times (-4) = -78 \text{ J} \).

05

Relating Work and Energy

The work-energy principle states that work done on an object results in a change in its kinetic energy. Here, the work done by gravity which is 78 J is transferred into increasing the kinetic energy of the mass as it falls.

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Key Concepts

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

Gravitational Force

Gravitational force is the force by which a planet or other celestial body attracts objects toward its center. It is crucial in the study of physics and is denoted by the formula: - \( F = m \cdot g \)This formula helps us determine the force of gravity acting on an object. Here, \( m \) stands for the mass of the object, and \( g \) is the acceleration due to gravity. On Earth, \( g \) is approximately \( 9.81 \text{ m/s}^2 \), a value derived from the planet's mass and radius. In this exercise, a mass of 2 kg experiences the gravitational force, calculated as:- \( F = 2.0 \times 9.81 = 19.62 \text{ N} \)Since gravity is a force always directed downward, its work on the object plays a central role in many phenomena, including free fall motion. As the object falls, the gravitational force propels it faster towards the Earth's surface.

Potential Energy

Potential energy, specifically gravitational potential energy (\( PE_G \)), is the energy an object possesses because of its position in a gravitational field. It's the energy that has the "potential" to do work when the object's position changes. The change in gravitational potential energy is calculated with:- \( \Delta PE_G = m \cdot g \cdot (h_f - h_i) \)Where \( h_i \) and \( h_f \) are the initial and final heights of the object. As this mass falls 4 meters, its potential energy decreases by the amount of work done by gravity, which is 78 J. This is expressed as:- \( \Delta PE_G = 19.62 \times (-4) = -78 \text{ J} \)This decrease represents the energy that gravitates into kinetic energy as the object moves downward. The conservation principle assures that the total energy remains constant, shifting forms, but not losing value.

Kinetic Energy

Kinetic energy (\( KE \)) is the energy an object possesses due to its motion. As the gravitational force does work on the falling object, its potential energy is converted to kinetic energy. The increase in kinetic energy can be quantitatively understood via the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.In this example, as the object falls:- The work done by gravity is 78 J, which directly corresponds to the increase in kinetic energy.Thus, the kinetic energy of the mass increases by 78 J as it descends, gaining speed. This principle not only explains energy transformation but also lays the foundation for solving numerous problems in physics, particularly in understanding motion dynamics and energy exchanges.