You drop a 2.00 kg book to a friend who stands on the...

You drop a $2.00 \mathrm{~kg}$ book to a friend who stands on the ground at distance $D=10.0 \mathrm{~m}$ below. If your friend's outstretched hands are at distance $d=1.50 \mathrm{~m}$ above the ground (Fig. $8-30$ ), (a) how much work $W_{g}$ does the gravitational force do on the book as it drops to her hands? (b) What is the change $\Delta U$ in the gravitational potential energy of the book-Earth system during the drop? If the gravitational potential energy $U$ of that system is taken to be zero at ground level, what is $U(\mathrm{c})$ when the book is released and (d) when it reaches her hands? Now take $U$ to be $100 \mathrm{~J}$ at ground level and again find (e) $W_{g}$, (f) $\Delta U,(\mathrm{~g}) U$ at the release point, and (h) $U$ at her hands.

You drop a $2.00 \mathrm{~kg}$ book to a friend who stands on the ground at distance $D=10.0 \mathrm{~m}$ below. If your friend's outstretched hands are at distance $d=1.50 \mathrm{~m}$ above the ground (Fig. $8-30$ ), (a) how much work $W_{g}$ does the gravitational force do on the book as it drops to her hands? (b) What is the change $\Delta U$ in the gravitational potential energy of the book-Earth system during the drop? If the gravitational potential energy $U$ of that system is taken to be zero at ground level, what is $U(\mathrm{c})$ when the book is released and (d) when it reaches her hands? Now take $U$ to be $100 \mathrm{~J}$ at ground level and again find (e) $W_{g}$,
(f) $\Delta U,(\mathrm{~g}) U$ at the release point, and
(h) $U$ at her hands.

Key Concepts

Work Done by Gravity

This concept refers to the energy transferred to or from an object when a force, such as gravity, causes displacement. For gravitational force, the work done is dependent on the change in height and the weight of the object. The sign of the work indicates whether energy is added to or taken from the object, making it crucial for understanding energy transformations in free-fall or drop scenarios.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in an object as a result of its vertical position in a gravitational field. It is calculated based on the mass of the object, gravitational acceleration, and its height relative to a chosen reference level. Understanding GPE is essential for analyzing how potential energy converts to kinetic energy during motion under gravity.

Reference Level for Potential Energy

The choice of reference level (or zero level) is arbitrary in gravitational potential energy calculations but important for consistency in problem-solving. Changing the zero point of potential energy does not affect the physics of energy conservation because energy differences remain constant. This concept is key when solving problems involving energy changes with different reference points.

Energy Conservation in Conservative Forces

This concept involves the principle that the total mechanical energy (the sum of kinetic and potential energies) in a system under conservative forces remains constant if no non-conservative work is done. In the context of gravitational problems, it explains how the work done by gravity is directly related to the change in gravitational potential energy, thereby linking dynamics and energy methods in problem solving.

Transcript

00:01 For this problem on the topic of conservation of energy, we are told that a 2 kilogram book is dropped to a person who stands on the ground a distance that is 10 meters below.

00:11 The person's hands are the distance of 1 .5 meters above the ground, as we can see in the figure, and we want to find various quantities relating to the energy of this motion.

00:23 So first of which is the amount of work that the gravitational force will do on the book as it drops to her hands.

00:31 Now we'll know that the vertical displacement is 10 meters minus 1 .5 meters which is 8 .5 meters downward.

00:39 So the work done by the gravitational force is mgd cosine phi, which is the mass of the book 2kg times the acceleration due to gravity 9 .8 meters per square second times a distance of 8 .5 meters and these are parallel to each other so that's multiplied by the cosine of zero degrees.

01:09 So gravity does work of 167 joules.

01:20 For part b, we want to find a change in gravitational potential energy of the book earth system during the drop.

01:32 Now one approach is to use the conservation of energy, and we can also calculate delta u where delta u is mgi with upward understood to be the positive y direction.

01:45 So delta u is equal to m g yf minus y i.

01:56 And so this change in gravitational potential energy is 2 kg times the acceleration due to gravity 9 .8 meters per square second times delta y, which is 1 .5 meters minus 10 meters.

02:16 So we get the change in gravitational potential energy of the system to be minus 167 joules.

02:24 The gravitational potential energy decreases and hence this is negative.

02:35 Now if the gravitational potential energy of the system is taken to be zero at ground level, we want to calculate the gravitational potential energy when the book is released.

02:46 So in part b we use the fact that ui is equal to m...

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