# 7.2 Kinetic energy and the work-energy theorem  (Page 3/7)

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## Determining the work to accelerate a package

Suppose that you push on the 30.0-kg package in [link] with a constant force of 120 N through a distance of 0.800 m, and that the opposing friction force averages 5.00 N.

(a) Calculate the net work done on the package. (b) Solve the same problem as in part (a), this time by finding the work done by each force that contributes to the net force.

Strategy and Concept for (a)

This is a motion in one dimension problem, because the downward force (from the weight of the package) and the normal force have equal magnitude and opposite direction, so that they cancel in calculating the net force, while the applied force, friction, and the displacement are all horizontal. (See [link] .) As expected, the net work is the net force times distance.

Solution for (a)

The net force is the push force minus friction, or ${F}_{\text{net}}\text{= 120 N – 5}\text{.}\text{00 N = 115 N}$ . Thus the net work is

$\begin{array}{lll}{W}_{\text{net}}& =& {F}_{\text{net}}d=\left(\text{115 N}\right)\left(\text{0.800 m}\right)\\ & =& \text{92.0 N}\cdot m=\text{92.0 J.}\end{array}$

Discussion for (a)

This value is the net work done on the package. The person actually does more work than this, because friction opposes the motion. Friction does negative work and removes some of the energy the person expends and converts it to thermal energy. The net work equals the sum of the work done by each individual force.

Strategy and Concept for (b)

The forces acting on the package are gravity, the normal force, the force of friction, and the applied force. The normal force and force of gravity are each perpendicular to the displacement, and therefore do no work.

Solution for (b)

The applied force does work.

$\begin{array}{lll}{W}_{\text{app}}& =& {F}_{\text{app}}d\phantom{\rule{0.25em}{0ex}}\text{cos}\left(0º\right)={F}_{\text{app}}d\\ & =& \left(\text{120 N}\right)\left(\text{0.800 m}\right)\\ & =& \text{96.0 J}\end{array}$

The friction force and displacement are in opposite directions, so that $\theta =\text{180º}$ , and the work done by friction is

$\begin{array}{lll}{W}_{\text{fr}}& =& {F}_{\text{fr}}d\phantom{\rule{0.25em}{0ex}}\text{cos}\left(\text{180º}\right)=-{F}_{\text{fr}}d\\ & =& -\left(\text{5.00 N}\right)\left(\text{0.800 m}\right)\\ & =& -\text{4.00 J.}\end{array}$

So the amounts of work done by gravity, by the normal force, by the applied force, and by friction are, respectively,

$\begin{array}{lll}{W}_{\text{gr}}& =& 0,\\ {W}_{N}& =& 0,\\ {W}_{\text{app}}& =& \text{96.0 J,}\\ {W}_{\text{fr}}& =& -\text{4.00 J.}\end{array}$

The total work done as the sum of the work done by each force is then seen to be

${W}_{\text{total}}={W}_{\text{gr}}+{W}_{N}+{W}_{\text{app}}+{W}_{\text{fr}}=\text{92}\text{.0 J}.$

Discussion for (b)

The calculated total work ${W}_{\text{total}}$ as the sum of the work by each force agrees, as expected, with the work ${W}_{\text{net}}$ done by the net force. The work done by a collection of forces acting on an object can be calculated by either approach.

## Determining speed from work and energy

Find the speed of the package in [link] at the end of the push, using work and energy concepts.

Strategy

Here the work-energy theorem can be used, because we have just calculated the net work, ${W}_{\text{net}}$ , and the initial kinetic energy, $\frac{1}{2}{m{v}_{0}}^{2}$ . These calculations allow us to find the final kinetic energy, $\frac{1}{2}{\text{mv}}^{2}$ , and thus the final speed $v$ .

Solution

The work-energy theorem in equation form is

${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}\text{.}$

Solving for $\frac{1}{2}{\text{mv}}^{2}$ gives

$\frac{1}{2}{\text{mv}}^{\text{2}}={W}_{\text{net}}+\frac{1}{2}{m{v}_{0}}^{2}\text{.}$

Thus,

$\frac{1}{2}{\text{mv}}^{2}=\text{92}\text{.}0 J+3\text{.}\text{75 J}=\text{95.}\text{75 J.}$

Solving for the final speed as requested and entering known values gives

$\begin{array}{lll}v& =& \sqrt{\frac{2\text{(95.75 J)}}{m}}=\sqrt{\frac{\text{191.5 kg}\cdot {m}^{2}{\text{/s}}^{2}}{\text{30.0 kg}}}\\ & =& \text{2.53 m/s.}\end{array}$

Discussion

Using work and energy, we not only arrive at an answer, we see that the final kinetic energy is the sum of the initial kinetic energy and the net work done on the package. This means that the work indeed adds to the energy of the package.

How submarines floats one water the same time sink in water
A submarine has the ability to float and sink. The ability to control buoyancy comes from the submarine'strim or ballast tanks which can be filled with either water or air, depending on whether the submarine needs to floator sink. When the submarine floats it means its trim tanks are filled with air
Arif
what is work
Force times distance
Karanja
product of force and distance...
Arif
Is physics a natural science?
what is the difference between a jet engine and a rocket engine.
explain the relationship between momentum and force
A moment is equivalent multiplied by the length passing through the point of reaction and that is perpendicular to the force
Karanja
How to find Squirrel frontal area from it's surface area?
how do we arrange the electronic configuration of elements
hi guys i am an elementary student
hi
Dancan
hello
are you an elementary student too?
benedict
no bro
yes
Che
hi
Miranwa
yes
Miranwa
welcome
Miranwa
what is the four equation of motion
Miranwa
what is strain?
SAMUEL
Change in dimension per unit dimension is called strain. Ex - Change in length per unit length l/L.
ABHIJIT
strain is the ratio of extension to length..=e/l...it has no unit because both are in meters and they cancel each other
How is it possible for one to drink a cold drink from a straw?
most possible as it is for you to drink your wine from your straw
Selina
state the law of conservation of energy
energy can neither be destroy or created,but can be change from one form to another
dare
yeah
Toheeb
it can neither be created nor destroyed
Toheeb
its so sample question dude
Muhsin
what is the difference between a principle and a law?
where are from you wendy .?
ghulam
philippines
Mary
why?
Mary
you are beautiful
ghulam
are you physics student
ghulam
laws are ment to be broken
Ge
hehe ghulam where r u from?
Muhsin
yes
dare
principle are meant to be followed
dare
south Africa
dare
here Nigeria
Toheeb
principle is a rule or law of nature, or the basic idea on how the laws of nature are applied.
Ayoka
Rules are meant to be broken while principals to be followed
Karanja
principle is a rule or law of nature, or the basic idea on how the laws of nature are applied.
tathir
what is momentum?
is the mass times velocity of an object
True
it is the product of mass and velocity of an object.
The momentum possessed by a body is generally defined as the product of its mass and velocity m×v
Usman
momentum is the product of the mass of a body of its velocity
Ugbesia
what about kg it is changing or not
no mass is the quantity or amount of body so it remains constant everywhere
Ahsan
yes
Siyanbola
remains constant
taha
mass of an object is always constant. and that is universally applied.
Shii
mass of a body never changes but the weight can change due to variance of gravity at different points of the world
Saheed
what is hookes law
Joshua
mass of an object does not change
SAMUEL
Is weight a scalar quantity
weight is actually a force of gravity with which earth attracts us downwards so it is a vector quantity. and it has both direction and magnitude
Ahsan
ty
Denise
weight is the earth pull of the body
Ugbesia
why does weight change but not mass?
Theo
Theo, the mass of an object can change but it depends on how you define that object. First, you need to know that mass is the amount of matter an object has, and weight is mass*gravity (the "force" that attracts object A to the object B mass).
Nicolas
So if you face object A with object B, you will get a different result than facing object A with object C, so the weight of object A changes but not its mass.
Nicolas
Now, if you have an object and you take a part away from it, you are changing it mass. Lets use the human body and fat loss process as an example.
Nicolas
When you lose weight by doing exercise, you are being attracted by the same object before and after losing weight so the change of weight is related to a change of mass not a change of gravity.
Nicolas
The explanation of this is simple, we are composed of smaller particles, which are itself objects, so the loose of mass of an object actually is the separation of one object is two different ones.
Nicolas
But if you define an object because of its form and characteristics and not the amount of mass, then the object is the same but you have taken a part of it mass away.
Nicolas
Theo, weight =mass. gravity, here mass is fixed everywhere but gravity change in different places so weight change not mass.
ABHIJIT
yup weight changes and mass does not. That's why we're 1/3 our weight on the moon
clifford
weight is the product of mass × velocity w=m×v = m(v-u) but v=u+1/2at^ weight is a scalar quantity mass of an obj is the amount of particles that obj cont
Usman
mass is fixed always while weight is dynamic
Usman
Why does water wet glass but mercury does not?
Yusuf
thanks guys
Theo
Yusuf Shuaibu, for water the Adhessive force between water molecules and glass is greater than the cohessive force between it's own molecules but for Mercury the cohessive force will be greater in comparison with adhessive force. For this water wet glass but Mercury does not.
ABHIJIT
in electrostatic e bonite rod electron is static. they cannot flow to other. because static. is it correct?
Is weight a scalar quantity
esther
wieght is the vector
ghulam
yes
Mohet
Yes
Karanja