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Work and Energy

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Work and Energy

- Work
– the product of force and the component of displacement in the direction of the force.

- work is a scalar quantity

- Without motion there is no work.

W = F ∙ d

The unit of work is the newton meter, which is called a joule (J)

(in honor of English Scientist James Prescott Joule)

- How much work is done on an object if a force of 30 Newtons [south] displaces the object 200 meters [south]?
- Solution:W = F∙d
= (30 N[S])(200 m [S])

= 6000 J

- Suppose force and displacement are not in the same direction.
- Work is defined to be the product of the force in the direction of the displacement and the displacement

F

W = (Fcosθ)∙d

d

θ

object

- As Alex pulls his red wagon down the sidewalk, the handle of the wagon makes an angle of 60 degrees with the pavement. If Alex exerts a force of 100 Newtons along the direction of the handle, how much work is done when the displacement of the wagon is 20 meters along the ground?

Power

- Power – the rate at which work is done
Power is also a scalar quantity, and its unit is Joules per second (J/s), also known as a watt(W).

- If 300 J of work is performed on an object in 1.0 minute, what is the power expended on the object?
P = W / t

P = 300J / 60 sec

P = 5 W

- A 200 N force is applied to an object that moves in the direction of the force. If the object travels with a constant velocity of 10m/s, calculate the power expended on the object.
P = 200N (10 m/s)

P = 2000 W

- Read pg. 80-83 Do # 1-25

Energy

Mechanical Energy

- Work done on an object changes its Kinetic Energy.
- Therefore, W = KEf – KEi = ΔKE.

- The formula for Kinetic Energy is
KE = ½ mv2

- A 10 kg object subjected to a 20. N force moves across a horizontal, frictionless surface in the direction of the force. Before the force was applied, the speed of the object was 2.0m/s. When the force is removed the object is traveling at 6.0 m/s. Calculate the following quantities: (a) KEi , (b) KEf , (c)ΔKE, (d) W, and (e) d.

(a)KEi = ½ mvi2

= ½ (10. kg)(2.0 m/s)2

= 20. J

(b)KEf = ½ mvf2

= ½ (10. kg)(6.0 m/s)2

= 180 J

(c) ΔKE = KEf – KEi

= 180 J – 20. J

= 160 J

(e) W = F∙d

d = W/F

= 160 J / 20. N

= 8.0 m

(d) W = ΔKE

= 160 J

- An object decreases its gravitational Potential Energy (PE) as it moves closer to the Earth
- To calculate the change in PE of an object we measure the work done on the object. The force needed to overcome gravity is Fg = mg. Therefore, since W = Fg ∙d , PE is defined as
ΔPE = mg Δ h

where Δh represents change in vertical displacement above the earth.

- A 2.00 kg mass is lifted to a height of 10.0 m above the surface of the Earth. Calculate the change in the PE of the object.
- Solution
ΔPE = mg Δ h

= (2.00 kg) (9.8 m/s2)(10.0m)

= 196 J

- For a change in gravitational energy to occur, there must be a change in the vertical displacement of an object; if it is moved only horizontally, the ΔPE = 0.
- If an object is moved up an inclined plane, its potential energy change is measured by calculating only its verticaldisplacement; the horizontal part does not change its PE

- Review book : read pg 84 Do # 26-38 (PE)
- Read pg 92-93 Do #57 -66 (KE)

Conservation of Mechanical Energy

- In a system, the sum of PE and KE (the total mechanical energy) is constant (i.e. conserved); a change in one is accompanied by an opposite change in the other.
ΔPE = -ΔKE

PEi + KEi = PEf +KEf

- A 0.50 kg ball is projected vertically and rises to a height of 2.0 meters above the ground. Calculate: (a) the increase in the ball’s PE
(b) the decrease in the ball’s KE

(c) the initial KE

(d) the initial speed of the ball

- The increase in the ball’s PE
ΔPE = mg Δ h

= (0.50 kg)(9.8 m/s2)(2.0 m)

= 9.8 J

(b) The decrease in the ball’s KE

ΔKE = -ΔPE

= -9.8 J

(c) The initial KE

- Recall that at its highest point, the speed of the ball is zero; therefore its KE is zero. So the initial KE represents the change in KE of the object .
ΔKE = KEf – KEi

-9.8J = 0 - KEi

KEi = 9.8 J

- The initial speed of the ball
KEi = ½ mvi2

Solving for vi = sqrt (2KEi /m)

= sqrt [ 2(9.8J)/(0.50kg) ]

= 6.3 m/s

Observe that the falling motion of the bob is accompanied by an increase in speed. As the bob loses height and PE, it gains speed and KE; yet the total of the two forms of mechanical energy is conserved

- A pendulum whose bob weighs 12 N is lifted a vertical height of 0.40 m from its equilibrium position. Calculate:
(a) change in PE between max height and equilibrium height

(b) Gain in KE and

(c) the velocity at the equilibrium point.

- Take PE at lowest point to be zero
ΔPE = mg Δ h = FgΔ h

= (12 N)(-0.40m)

= -4.8 J

(b) ΔKE = -ΔPE

= -(-4.8 J)

= 4.8 J

( c ) First must calculate mass of bob

Fg = mg

m = Fg / g

= 12N / 9.8m/s2

= 1.2 kg

Then calculate velocity

KE= ½ mv2

v = sqrt (2KE/m)

= sqrt [ 2(4.8J)/(1.2kg) ]

= 2.8 m/s

Elastic Potential Energy and Springs

- The English scientist Robert Hooke was able to show that the magnitude of a force (F) is directly proportional to the elongation (stretch) of compression of a spring (x) within certain limits.
Fs = kx

k – spring constant – unit is the newton per meter (N/m)

Note: the greater the constant, the stiffer the spring

Spring is attached to a wall. If a force is applied and stretch the string to the right, work has been done. This work has been converted into the spring’s potential energy

– Elastic Potential Energy

PEs = ½ kx2

Area under the graph equals the work done.

Fs = kx

W

- A spring whose constant is 2.0 N/m is stretched 0.40 m from its equilibrium position. What is the increase in the elastic potential energy of the spring?
Solution

PEs = ½ kx2

= ½ (2.0 N/m ) (0.40 m)2

= 0.16 J

- In an elastic collision BOTH kinetic energy and momentum are conserved
p1i + p2i = p1f + p2f

KE1i+ KE2i = KE1f +KE2f

p1i

p2i

p1f

p2f

1

2

1

2

1

2

KE1i

KE2i

KE2f

KE1f

- In inelastic collisions, the kinetic energy that is “lost” is converted into internal energy (Q) of the objects by frictional forces.
- These systems are called nonideal mechanical systems.
- The energy is constant
ET = PE + KE + Q

A change in the internal energy of an object is usually accompanied by a change in temperature

Simple Machines and Work

- A simple machine is a device that allows work to be done and offers and advantage to the user. Ex: pulleys, levers, inclined planes, wheels and axles and screwdrivers
Win = Wout

Fin ∙ din = Fout ∙ dout

Fout / Fin = din / dout

Mechanical Advantage (MA): Fout / Fin

Ideal MA (IMA) assumes no friction (use din / dout )

Actual MA (AMA) is always less than IMA (use Fout / Fin )

The efficiency of machine is AMA/IMA ratio and this value is always less than 100%

- A 100 N object is moved 2 m up an inclined plane whose end is lifted 0.5 m from the floor. If a force of 50 N is needed to accomplish this task, calculate the (a) IMA
(b) AMA

And (c) efficiency of the inclined plane

Input force (Fin ) = 50 N (force needed to move object)

Output force (Fout ) = 100N (force that has been lifted)

Input distance (din ) = 2m (distance moved along plane)

Output distance (dout ) = 0.5m (distance object is raised)

- IMA = din / dout = 2m / 0.5m = 4
- AMA = Fout / Fin = 100N / 50 N = 2
- efficiency = AMA/IMA = 2/4 = 0.5 (50%)

Internal Energy and Work

- Internal Energy of a system is the total kinetic and potential energies of the atoms and molecules that make up the system.
- Recall: a change in the internal energy of an object is usually accompanied by a change in its temperature.

- Force is used to move an object along a horizontal table at constant speed.
- Has work been done?
- Is there a change in Kinetic Energy? Why or why not?
- Is there a change in Potential Energy? Why or why not?
- How was work used?

Yes W = F ∙ d

No – speed is constant

No – table is horizontal - so no change in height

Used to overcome friction between object and table so internal energy of the object-table system has been increased by work done.

The Laws of Thermodynamics

The study of the relationships among heat, work, and energy in the universe

- Energy can neither be created nor destroyed. It can only change forms.
- The change in the internal energy of a system (ΔU) is equal to the heat (Q) that the system absorbs (or releases) minus the work (W) it does (or has done on it)
ΔU = Q – W

- Result if the work of the French physicist Nicolas Carnot with heat engines.
- Law states heat cannot flow from colder object to a warmer one without work being done on the system.
- Ex. Refrigerators must be run by motors in order to withdraw heat from objects placed in them

- No heat engine can be 100% efficient. Some of the heat absorbed by the engine must be lost in the random motion of its molecules.
- Entropy is the measure of this disorder

- As temperature approaches absolute zero, (0 K) the entropy of a system approaches a constant minimum
- The efficiency of a heat engine depends on its operating temperatures; engine would reach 100 % efficiency only at 0 K.
- Engine cannot be completely efficient, therefore 0K cannot be reached.