# GIVE THE TOTAL MECHANICAL ENERGY OF THE BALL E IN TERMS OF MAXIMUM HEIGHT HM IT REACHES

Exordinary gravitational potential power in terms of occupational done versus gravity.Sexactly how that the gravitational potential energy of an object of mass m at height h on Planet is given by PEg = mgh.Sexactly how exactly how understanding of the potential energy as a role of position have the right to be provided to simplify calculations and also describe physical sensations.

## Work Done Against Gravity

Climbing stairs and lifting objects is occupational in both the scientific and also day-to-day sense—it is occupational done versus the gravitational force. When there is work-related, tbelow is a change of energy. The work done versus the gravitational force goes right into a crucial form of stored energy that we will certainly check out in this area.

Figure 1. (a) The work-related done to lift the weight is stored in the mass-Planet system as gravitational potential power. (b) As the weight moves downward, this gravitational potential power is transferred to the cuckoo clock.

Let us calculate the job-related done in lifting an object of mass m with a height h, such as in Figure 1. If the object is lifted directly up at constant speed, then the pressure required to lift it is equal to its weight mg. The work-related done on the mass is then WFdmgh. We define this to be the gravitational potential energy (PEg) put right into (or acquired by) the object-Earth device. This power is linked through the state of separation in between two objects that tempt each other by the gravitational force. For convenience, we refer to this as the PEg got by the object, recognizing that this is energy stored in the gravitational field of Earth. Why carry out we use the word “system”? Potential power is a residential property of a system fairly than of a single object—as a result of its physical place. An object’s gravitational potential is because of its position relative to the surroundings within the Earth-object device. The force applied to the object is an exterior pressure, from outside the device. When it does positive occupational it rises the gravitational potential energy of the device. Because gravitational potential energy relies on family member position, we need a recommendation level at which to collection the potential energy equal to 0. We commonly choose this suggest to be Earth’s surface, however this suggest is arbitrary; what is essential is the difference in gravitational potential energy, because this difference is what relates to the work done. The difference in gravitational potential power of a things (in the Earth-object system) between two rungs of a ladder will certainly be the exact same for the initially 2 rungs as for the last two rungs.

## Converting Between Potential Energy and Kinetic Energy

Gravitational potential energy might be converted to various other forms of energy, such as kinetic power. If we release the mass, gravitational pressure will certainly perform an amount of work-related equal to mgh on it, thereby boosting its kinetic power by that very same amount (by the work-energy theorem). We will uncover it more helpful to take into consideration just the convariation of PEg to KE without clearly considering the intermediate step of occupational. (See Example 2.) This shortcut renders it is easier to settle problems using energy (if possible) rather than clearly utilizing forces.

More specifically, we specify the change in gravitational potential power ΔPEg to be ΔPEg = mgh, wright here, for simplicity, we denote the change in height by h rather than the usual Δh. Keep in mind that h is positive as soon as the final height is greater than the initial elevation, and vice versa. For instance, if a 0.500-kg mass hung from a cuckoo clock is raised 1.00 m, then its change in gravitational potential energy is

eginarraylllmgh&=&(0.500 ext kg)(9.80 ext m/s^2)(1.00 ext m)\ ext &=&4.90 ext kgcdot extm^2 ext/s^2=4.90 ext Jendarray\

Keep in mind that the units of gravitational potential power rotate out to be joules, the same as for work-related and other forms of power. As the clock runs, the mass is lowered. We have the right to think of the mass as progressively providing up its 4.90 J of gravitational potential power, without directly considering the pressure of gravity that does the work.

## Using Potential Energy to Simplify Calculations

Figure 2. The change in gravitational potential power (ΔPEg) in between points A and also B is independent of the path.

The equation ΔPEg = mgh uses for any kind of path that has a readjust in height of h, not just once the mass is lifted directly up. (See Figure 2.) It is much much easier to calculate mgh (a simple multiplication) than it is to calculate the job-related done alengthy a complicated path. The idea of gravitational potential energy has actually the double advantage that it is very generally applicable and it makes calculations much easier.

From currently on, we will think about that any kind of adjust in vertical position h of a mass m is accompanied by a adjust in gravitational potential energy mgh, and also we will avoid the tantamount but even more hard task of calculating work done by or against the gravitational pressure.

ΔPEg = mgh for any kind of path in between the two points. Gravity is among a small course of forces wright here the occupational done by or versus the pressure depends just on the starting and finishing points, not on the course between them.

### Example 1. The Force to Stop Falling

A 60.0-kg perboy jumps onto the floor from a elevation of 3.00 m. If he lands stiffly (with his knee joints compushing by 0.500 cm), calculate the pressure on the knee joints.

Strategy

This person’s power is lugged to zero in this case by the work done on him by the floor as he stops. The initial PEg is transformed right into KE as he drops. The work-related done by the floor reduces this kinetic energy to zero.

Solution

The occupational done on the perkid by the floor as he stops is given by WFd cos θ = −Fd, via a minus sign bereason the displacement while protecting against and the force from floor are in oppowebsite directions (cos θ = cos 180º = −1). The floor clears energy from the device, so it does negative work-related.

The kinetic power the perchild has upon reaching the floor is the amount of potential energy shed by falling through height h: KE = −ΔPEg = −mgh.

The distance d that the person’s knees bfinish is a lot smaller than the elevation h of the autumn, so the added readjust in gravitational potential power in the time of the knee bend is ignored.

The work-related W done by the floor on the person stops the person and brings the person’s kinetic power to zero: W = −KE = mgh.

Combining this equation through the expression for W gives −Fdmgh.

Recalling that h is negative because the person dropped down, the pressure on the knee joints is offered by

displaystyleF=-fracmghd=-fracleft(60.0 ext kg ight)left(9.80 ext m/s^2 ight)left(-3.00 ext m ight)5.00 imes10^-3 ext m=3.53 imes10^5 ext N\

Discussion

Such a large pressure (500 times even more than the person’s weight) over the brief impact time is enough to break bones. A much better means to cushion the shock is by bending the legs or rolling on the ground, increasing the moment over which the pressure acts. A bending activity of 0.5 m this method returns a pressure 100 times smaller than in the example. A kangaroo’s hopping shows this approach in action. The kangaroo is the just big pet to use hopping for locomotion, but the shock in hopping is cushioned by the bending of its hind legs in each jump. (See Figure 3.)

Figure 3. The occupational done by the ground upon the kangaroo reduces its kinetic power to zero as it lands. However before, by using the force of the ground on the hind legs over a longer distance, the affect on the bones is diminished. (credit: Chris Samuel, Flickr)

### Example 2. Finding the Speed of a Roller Coaster from its Height

What is the last rate of the roller coaster displayed in Figure 4 if it starts from rest at the optimal of the 20.0 m hill and work done by frictional pressures is negligible?What is its final speed (aacquire assuming negligible friction) if its initial rate is 5.00 m/s?

Figure 4. The rate of a roller coaster increases as gravity pulls it downhill and also is biggest at its lowest point. Viewed in regards to power, the roller-coaster-Planet system’s gravitational potential energy is converted to kinetic power. If occupational done by friction is negligible, all ΔPEg is converted to KE.

Strategy

The roller coaster loses potential power as it goes downhill. We neglect friction, so that the continuing to be pressure exerted by the track is the normal force, which is perpendicular to the direction of movement and does no work. The net occupational on the roller coaster is then done by gravity alone. The loss of gravitational potential energy from relocating downward with a distance h equals the gain in kinetic power. This can be composed in equation develop as −ΔPEg = ΔKE. Using the equations for PEg and KE, we deserve to solve for the last speed v, which is the desired amount.

Systems for Part 1

Here the initial kinetic energy is zero, so that Delta extKE=frac12mv^2\. The equation for adjust in potential power says that ΔPEg = mgh. Due to the fact that h is negative in this case, we will certainly recompose this as ΔPEg = −mg|h| to present the minus sign plainly. Hence, −ΔPEg = ΔKE becomes mg|h|=frac12mv^2\.

Solving for v, we find that mass cancels and that v=sqrt\.

Substituting well-known values,

eginarraylllv&=&sqrt2left(9.80 ext m/s^2 ight)left(20.0 ext m ight)\ ext &=&19.8 ext m/sendarray\

Systems for Part 2

Aget −ΔPEg = ΔKE. In this situation tbelow is initial kinetic power, so

Delta extKE=frac12mv^2-frac12mv_0^2\.

Hence, mg|h|=frac12mv^2-frac12mv_0^2\.

Rearvarying offers frac12mv^2=mg|h|+frac12mv+0^2\.

This indicates that the final kinetic energy is the amount of the initial kinetic power and also the gravitational potential energy. Mass aget cancels, and v=sqrt+v_0^2\.

This equation is exceptionally similar to the kinematics equation v=sqrtv_0^2+2ad\, yet it is more general—the kinematics equation is valid just for consistent acceleration, whereas our equation above is valid for any type of route regardmuch less of whether the object moves through a consistent acceleration. Now, substituting well-known worths gives

eginarraylllv&=&sqrt2left(9.80 ext m/s^2 ight)left(20.0 ext m ight)+left(5.00 ext m/s ight)^2\ ext &=&20.4 ext m/sendarray\

Discussion and Implications

First, note that mass cancels. This is quite continual with observations made in Falling Objects that all objects loss at the very same price if friction is negligible. Second, only the speed of the roller coaster is considered; there is no indevelopment about its direction at any type of suggest. This reveals another basic truth. When friction is negligible, the rate of a falling body counts just on its initial speed and height, and also not on its mass or the route taken. For example, the roller coaster will certainly have actually the very same last rate whether it drops 20.0 m right down or takes a more complex route choose the one in the figure. Third, and also maybe unexpectedly, the last speed in component 2 is better than in part 1, however by much much less than 5.00 m/s. Finally, note that rate can be uncovered at any height alengthy the way by simply utilizing the proper value of h at the point of interemainder.

We have viewed that work done by or versus the gravitational pressure relies only on the founding and also ending points, and not on the course in between, permitting us to define the simplifying idea of gravitational potential energy. We deserve to perform the very same thing for a couple of various other pressures, and we will check out that this leads to a formal definition of the regulation of conservation of power.

### Making Connections: Take-Home Investigation—Converting Potential to Kinetic Energy

One have the right to research the conversion of gravitational potential power into kinetic energy in this experiment. On a smooth, level surface, usage a ruler of the type that has actually a groove running alengthy its length and a book to make an incline (see Figure 5). Place a marble at the 10-cm place on the ruler and let it roll down the leader. When it hits the level surconfront, meacertain the moment it takes to roll one meter. Now place the marble at the 20-cm and also the 30-cm positions and aget measure the times it takes to roll 1 m on the level surconfront. Find the velocity of the marble on the level surface for all 3 positions. Plot velocity squared versus the distance traveled by the marble. What is the form of each plot? If the shape is a directly line, the plot reflects that the marble’s kinetic energy at the bottom is proportional to its potential energy at the release allude.

Figure 5. A marble rolls dvery own a ruler, and also its rate on the level surchallenge is measured.

## Section Summary

Work done versus gravity in lifting an item becomes potential power of the object-Planet device.The readjust in gravitational potential energy, ΔPEg, is ΔPEg = mgh, with h being the increase in elevation and also g the acceleration as a result of gravity.The gravitational potential energy of an object close to Earth’s surface is as a result of its position in the mass-Earth device. Only differences in gravitational potential energy, ΔPEg, have actually physical definition.As an object descends without friction, its gravitational potential power alters into kinetic power equivalent to raising rate, so that ΔKE = −ΔPEg

### Conceptual Questions

In Example 2, we calculated the final speed of a roller coaster that descfinished 20 m in height and had an initial rate of 5 m/s downhill. Suppose the roller coaster had actually had actually an initial speed of 5 m/s uphill instead, and also it coasted uphill, quit, and then rolled earlier dvery own to a last suggest 20 m below the begin. We would certainly uncover in that situation that it had actually the exact same last rate. Exsimple in regards to conservation of energy.Does the work-related you carry out on a book when you lift it onto a shelf depend on the route taken? On the moment taken? On the height of the shelf? On the mass of the book?

### Problems & Exercises

A hydroelectrical power facility (view Figure 6) converts the gravitational potential power of water behind a dam to electrical power. (a) What is the gravitational potential energy loved one to the generators of a lake of volume 50.0 km3 (mass = 5.00 × 1013 kg), provided that the lake has actually an average elevation of 40.0 m above the generators? (b) Compare this via the energy stored in a 9-megaton fusion bomb.

Figure 6. Hydroelectric facility (credit: Denis Belevich, Wikimedia Commons)

(a) How much gravitational potential power (relative to the ground on which it is built) is stored in the Great Pyramid of Cheops, provided that its mass is about 7 × 109 kg and its center of mass is 36.5 m above the neighboring ground? (b) How does this power compare through the everyday food intake of a person?Suppose a 350-g kookaburra (a big kingfisher bird) picks up a 75-g snake and raises it 2.5 m from the ground to a branch. (a) How much work did the bird perform on the snake? (b) How a lot work-related did it do to raise its own facility of mass to the branch?In Example 2, we uncovered that the speed of a roller coaster that had descended 20.0 m was just slightly better once it had actually an initial speed of 5.00 m/s than when it started from rest. This suggests that ΔPE >> KEi. Confirm this statement by taking the ratio of ΔPE to KEi. (Keep in mind that mass cancels.)A 100-g toy auto is driven by a compressed spring that starts it moving. The vehicle follows the curved track in Figure 7. Sexactly how that the last speed of the toy auto is 0.687 m/s if its initial rate is 2.00 m/s and it coasts up the frictionmuch less slope, obtaining 0.180 m in altitude.

Figure 7. A toy auto moves up a sloped track. (credit: Leszek Leszczynski, Flickr)

In a downhill ski race, surprisingly, bit advantage is gained by gaining a running start. (This is bereason the initial kinetic energy is tiny compared via the acquire in gravitational potential power on also little hills.) To show this, find the last speed and also the moment taken for a skier who skies 70.0 m along a 30º slope neglecting friction: (a) Starting from remainder. (b) Starting via an initial rate of 2.50 m/s. (c) Does the answer surpincrease you? Discuss why it is still advantageous to acquire a running begin in exceptionally competitive events.

## Glossary

gravitational potential energy: the energy an item has because of its place in a gravitational field

### Schosen Solutions to Problems & Exercises

1. (a) 1.96 × 1016 J; (b) The ratio of gravitational potential power in the lake to the power stored in the bomb is 0.52. That is, the energy stored in the lake is about fifty percent that in a 9-megaton fusion bomb.

3. (a) 1.8 J; (b) 8.6 J

5. v_f=sqrt2gh+v_0^2=sqrt2left(9.80 ext m/s^2 ight)left(-0.180 ext m ight)+left(2.00 ext m/s ight)^2=0.687 ext m/s\