Work and Energy Unit: Worksheet 1, Work and Power Calculations

Work is the product of a Force vector x a displacement vector. W F·~x. Be careful, the force has to be directed along the displacement vector. If the force is at an angle, you will have to use cosine theta to get the correct force vector. Work is measured in a unit called, "Joules" or J for short.

Power = (Energy or Work) / time. Power is measured in Watts (W). One watt is just one Joule/sec

1. A 60 kg box is lifted by a rope a distance of 10 meters straight up at constant speed.

How much work is required to lift the box? (answer 5880 J)

How much power is required to complete this task in 5 seconds? (answer = 1176 W)

2. Hulky and Bulky are two workers being considered for ,a job at the UPS loading dock. Hulky boasts that he can lift a 100 kg box 2.0 meters vertically, in 3.0 seconds. Bulky counters with his claim of lifting a 200 kg box 5.0 meters vertically, in 20 seconds.

How much work was required to lift the boxes? (Hulky's box requires 1960 J of work and Bulky's box requires 9800 Joules of work.)

Which worker has a greater power rating? (answer = Hulky, 653 W> 490 W)

Unit 10: work and energy, worksheet 1, p. 1 2112/2009

Diagram A Diagram B Diagram C

F=100 N ~ 15,kg

F=Z B······~~t.....

15 kg

IF 15 kg

A 100 N force is applied to move a 15 kg object a horizontal distance of 5 meters at constant speed,

A 100 N force is applied at an angle of 300 to the horizontal to move a 15 kg object at a constant speed for a horizontal distance of 5 m.

An up'w'ard force is applied to lift a 1 5 kg object to a hei ght of 5 mete rs at constant speed.

Draw a force diagram for each diagram.

Find the work of the applied force on each of the three diagrams.

Find the power of the applied force if it takes 5 seconds for each motion to occur.

Diagram A

(answers = work 500 J and the power = 100 W)

DiagramB

(answers = work = 433 J and the power 86.6 W)

Diagram C

(answers = work = 735 J and the power = 147 W)

Unit 10: work and energy, worksheet 1, p. 2 2/12/2009

Work and Energy Unit: Worksheet 2: Using Pie Graphs to map Energy

Use pie charts to analyze the energy changes in each situation given.

• Carefully label the pies to correspond with the positions of the objects given. (A, B, C, etc.) • The pie graphs should be divided where needed and labeled with the energy storage mechanisms

involved. Use elastic energy, gravitational potential energy, kinetic energy, and dissipated energy.

1. A ball is held above the ground, and then is dropped so it falls straight down. Do a pie chart for each position. Neglect any air resistance and dissipative energy (no friction) for this problem. (Restrict your analysis to the ball in the air, BEFORE it hits the ground.)

o'0,0 o ~o o

2. A wind-up toy is wound up, then "walks" across a table and comes to a stop as the spring winds down. Do a pie chart for each position. Into what type of energy does turn?

3. A baseball is thrown up in the air and then falls back down. Place velocity vectors beside each image of the baseball in the drawing, and do a pie chart for each position. Neglect any air resistance and dissipative energy (no friction). (Restrict your analysis to the ball in the air, BEFORE it hits the ground.) Use the ground as the reference line for Eg•

Work and energy unit: worksheet 2 p. I 2112/2009

III •

4. A ball rolls to a stop on the floor. •Do a pie chart for each position. This time, include friction (acts as a dissipative energy).

5. An object rests on a coiled spring, and is then launched upwards. Do a pie chart for each position. Neglect any air resistance and other frictional effects (no dissipative energy) . .~

'ii

6. A truck is driven at constant speed down the street. Do a pie chart for each position. Include friction which acts as a dissipative energy.

p

5tUL f) L

Work and energy unit: worksheet 2 p.2 2112/2009

Work and Energy Unit: Worksheet 3 Spring Constants and Elastic Energy

1. The following picture looks like the setup you had in your lab.

Let's say that you attach a spring and pull on it using the meterstick to guide you. The meterstick readings and your elastic forces are given in the following data table.

Meterstick reading Your Elastic Force (m) (delta x) (Reading on

the spring scale (N))

0.00 0 0.20 10 0.40 20 0.60 30 0.80 40 1.00 50

First graph the data points and obtain the slope. This slope is the spring constant (k). This is Hooke's Law: Fet = k(8X)

Elastic Force (Fel in N) 50r~"~O~~",_~"~N"~,~'~~~~-~=~"~W~~~'4

o delta x in meters o

What did you get for your slope? This is 'k', the spring constant.

2. Using your k and the elastic energy formula: Eelastic == 12 k*(L1x2), determine the energy of

the spring when you pull it back the following distances:

a) 0.5 m

b) 1.0 m

What is the ratio of these two energies? Why?

Energy and Work Unit: worksheet 3 p. 1 adg

Next problem

Suppose in the lab, one group found that Fel = 1000 N/nl*(Llx). Construct a graph of force vs. displacement (LlX) on the graph provided and use the graph to help you answer the next two questions.

Elastic Force (F el in N) 250f~-~~~~~~~~~~M~~~~~~~'-~~~~~

1. Graphically determine the amount of energy stored while stretching the spring described above from x = ato x = 0.10 m. (answer 5J)

2. Graphically determine the amount of energy stored while stretching the spring described above from x = 0.15 to x = 0.25 m. (answer = 20 J)

oI...--_..I...-_-'--_--'-_--'i._---'...... delta x in meters o 0.25

The graph at left was made from data collected during an investigation of the relationship between the amount two different springs stretched, when different forces were applied.

3. F or each spring determine the spring constant. (answers: kspringl 5 N/m, kspring2 = 8 N/m)

4. For each spring, compare a) the amount of force required to stretch

the spring 3.0 m. (answers: Fspringl 15 N, F spring2 = 24 N)

b) the Eel stored in each spring when stretched 3.0 m. (answers: Ee] (spring])

• I 22.5 J, Eel (spring2) = 36 J 0.00 2.00 4.00 6.00 8.00

Ax(m)

36.0­

32.0 ­

28.0 ­

24.0 ­

~20.0 :

Q)

e -~16.0 -

12.0 ­

8.00 ­

4.00 ­

o Data Set 1 ~ Data Set 2

0.00 -. . I • • • I • • • I ••• I

5. Determine the amount that spring 2 needs to be stretched in order to store 24 joules of energy. (answer = 2.45 m)

Energy and Work Unit: worksheet 3 p.2 adg

Work and Energy Unit: Worksheet 4, Energy and Work Calculations

Use the three equations for energy in your calculations:

1 Kinetic Energy Energy of Motion = Ek = -m(v2

) Measured in Joules (J) 2

Elastic Energy Energy stored in Elastic Materials = Eel = -1

k(Ax2 ) Measured in Joules (J)

2

Gravitational Potential Energy = Energy of an object at a height in a gravity field = Eg mgh

Measured in Joules (J)

Physical Work = Work I put into an object = External Force * distance along which the force acts = F(d) Measured in Joules (J)

1. A ball has a mass of 0.05 kg and is thrown at 30 mls upward. What is its kinetic energy? (answer = 22.5 J)

2. What is the speed of a 1.5 kg rock falling with a kinetic energy of 48 J? (answer 8 mls)

3. After the rock in question 2 hits the ground, a student lifts it straight up two meters. Now what is its gravitational potential energy? (answer 29.4 J)

4. Earlier this morning, my 61.23 kg body falls 0.5 meters out of bed. How much gravitational potential energy did I lose? (answer = 300 J of energy lost or -300 J)

5. A 0.04 kg rubber ball drops from a height of 5 m to the ground and bounces back to a height of 3 meters. (a) How much gravitational potential energy does the ball lose on the trip down? (answer 1.96 J)

(b) How much gravitational potential energy does the ball regain on the trip back up? (answer = 1.18 J)

(c) What fraction of energy was lost and where did this energy go? (answer = 40%, dissipated energy, actually it goes to thermal energy and slightly increases the temp of the ball)

Work and Energy Unit: Worksheet 4 p. 1 adg, 2/12/2009

Worksheet 4: Energy and Work Calculations (continued)

6. A catapult propels a golfball upward. To launch the golfball, a rubber band with a spring constant (k) of 10 N/m is pulled back 0.2 meters. Determine how much elastic energy was stored in the rubber band to launch the golfball. (Answer 0.2 J)

7. Your car runs out of gas and you get out and push it. If you apply a force of 800 N for 200 m (the distance to the nearest gas station), how much work did you do to push it? (answer 160,000 J)

8. A camper uses a rope and a pail to get water from a well. If the pail with water has a mass of20 kg and the camper lifts it 3.5 m, how much work was done to lift the pail up? (answer

686 J)

9. A 2.0 kg puck accelerated at 5 m/s2 for 0.5 m across a 'frictionless' air hockey table. How much work was done on the puck to accelerate it? (use F n1a to help you and then Fd = Work, answer = 5 J)

10. A 0.5 kg rubber ball is thrown straight up into the air. At a height of 20 m above the ground, it is traveling at 15 m/s. (a) What is the ball's kinetic energy at this point? (answer = 56.25 J)

(b) What is its gravitational potential energy relative to the ground? (answer 98 1)

(c) If all of the Eg the ball has in part b was originally Ek at the ground, tellll1e how fast the ball was traveling initially when it was thrown up in the air. (answer 24.8 m/s)

11. A pendulum is pulled back so it is 30 cnl (0.30 m) higher than a reference line. If all of the Eg is turned to Ek at the bottom of the swing, calculate its velocity at the bottom. (answer 2.4 m/s)

Work and Energy Unit: Worksheet 4 p.2 adg, 10/25/2009

5

m =20 kg

0"7­

Work and Energy Unit: Worksheet 5 Quantitative Bar Graphs and Problems For each situation shown below:

1. Assume the systems to be frictionless, unless stated otherwise. Also, assume there is no air resistance. 2. Complete the energy bar graphs QUANTITATIVELY (numerically accurate using your energy equations) putting in your own

scale. 3. In the space below each diagram use conservation of energy equations to solve for the quantity called for in the question.

1. A moving cart hits a spring, traveling at 5.0 m/s at Energy Bar Graphs the time of contact. At the instant the cart is Initial Final motionless, by how much displacement is the Ek Eg Eel Eg Eel Ediss spring compressed (Ax)? (answer = 2 m)

m= B.O kg k = 5011 v= 5_0 mls ry==-O m

wio Finallritial OJ

2. Determine the final velocity of the cart if the cart starts at 5 m above the ground. (answer = 9.9 m/s) Initial Final

Ek Eel Ek Eg Eel Ediss

v=O

5

m = 20 kg

0-­OJ

3. Determine the final velocity of the cart, assuming that 10% of the energy is dissipated by friction. It again starts 5 meters above the ground. (answer = 9.4 m/s)

Initial Final Eel Eg Ediss

v=O

v=? ......................................................................................................................................................................................................................

_-.w...w...._....................................................................................................................................................................................................................

OJ Work and Energy Unit: worksheet 5 p. 1 adg, 2112/2009

4. A block of mass 0.5 kg is placed on a spring, (spring constant, k = 100 N/m) compressing it 0.30m. What height does the block above its initial position when launched by the spring? (answer 0.92 m)

Initial Final Ek Eg Eel Ek Eg Eel Ediss

~v=O

OJ Initial Final

5. The bullet with only Ek strikes a block ofwood that exerts, on average, a force of 50,000N opposing the nlotion of the bullet. How far does the bullet penetrate? (answer = 0.031 m or 3.1 cm)

Initial Final Eg Ek Eg Eel Edissm = 25 g (0.025 kg) '1= 0

y= 350 mls

-=-­ FI Initial Final

OJ

6. A crate is propelled up a hill by a tightly coiled spring. The height of the hill is 1 m. The spring constant is 200 N/m and the ~x 0.5 m. The mass of the crate is 1 kg. If 20% of the spring energy is dissipated by the friction of the surface as the crate moves along it upward, determine the final velocity of the crate. (answer = 4.52 m/s)

Initial Final Ek Eg Eel Ek Eg

y v>O

Initial OJ

0"';""

Work and Energy Unit: worksheet 5 p.2 2112/2009

Work and Energy Unit: Worksheet 6, Conservation of Energy Calculations All types of Energy before All types of Energy after

Mechanical energy = Ek +

1. A 56 kg cliff diver run to the edge of a cliff and dives off the edge into the water 4 m below. If she is moving at 8 mls horizontally the instant she leaves the cliff, first make a drawing and then determine the following:

(a) Her gravitational potential energy relative to surface of the water when she leaves the cliff (answer = 2195.2 J)

(b) What is her kinetic energy when she leaves the cliff? (answer 1792 J)

(c) The addition of (a) and (b) [Eg + Ek] is equal to "MECHANICAL ENERGY." What is the diver's mechanical energy when she leaves the cliff? (answer = 3987.2J)

(d) Consequently, what would be her mechanical energy just before she enters the water? (answer 3987.2 J)

(e) Knowing this, you should be able to calculate her speed as she enters the water (v = 11.93 mls).

2. An arrow is shot at the apple atop the head of Walter, William Tell's son. The arrow is 0.2 kg, is moving at 20 mls when it leaves the arrow, and is shot 1.5 meters above the ground.

(a) Find the total mechanical energy of the arrow as it leaves the bow. (answer = 42.94 J)

(b) How high above the ground does the arrow strike the tree if it IS

moving at 10 mls at the instant it strikes the tree? (answer 16.81 m)

Work and energy unit: worksheet 6 p. 1 adg, 2/12/2009

3. A 300 kg snowmobile (total mass with the driver) is traveling at 16 mls when it comes to the edge of a small cliff that is 2.5 m above the ground. The driver drives over the cliff and falls parabolically to the snow on the ground without changing speed.

(a) What is the total mechanical energy of the driver and snowmobile as he 45,750 J).

(b) How fast is the snowmobile going when it lands on the ground? (answer 17.46 mls)

(a) How high does Casey get above the ground in his swing? (answer = 3.27 m above the ground)

(b) Ryan, who is 20 kg, gives it a tree and reaches a height of3 meters above the ground? How fast was Ryan running to achieve this height? (answer 7.7 m/s)

4. Casey and Ryan, pretending that they are playing in the jungle, suspend a rope from an overhead tree limb. Casey, of mass, 40 kg, runs horizontally at 8 mls along the flat ground, grabs the rope, and swings off the level n",~',," .... r1

5. An amusement park has a slide for which participants are given a cloth sack to sit on. The top of the slide is 6 m high.

(a) Determine the speed attained at the bottom of the slide by a 30 kg child. Assume that the child starts from rest and that the slide is frictionless. (answer = 10.84 mls)

(b) Repeat your calculation for ( a) but this time assume that 80% of the gravitational potential energy that the child has at the top of the slide is required to overcome friction, and is therefore dissipated energy (lost as heat) (answer = 4.85 mls)

Work and energy unit: worksheet 6 p.2 adg, 2/12/2009

Work and Energy Unit: Worksheet 7, Energy Conservation Problems

Start each solution with a force diagram.

1. A baseball (m 140 g or 0.14 kg) traveling at 30 mls moves a fielder's glove backward 35 cm (0.35 m) when the ball is caught. a. What is the ball's Ek before it hits the fielder's mitt? (answer 63 J)

b. What was the average force exerted by the ball on the glove? (answer = -180 N)

2. A 60 kg student jumps fronl the 10 meter platform into the swimming pool below. a. Determine her Eg at the top of the platform. (answer = 5880 J)

b. How much Ek does she possess at impact? What is her velocity at impact? (neglect air resistance) (answer = 5880 J at impact, v 14 mls)

c. If she jumped from a platform that was twice as high, how many times greater would be her

velocity at impact? (answer = 1.41 or .J2 times greater)

d. How much higher would the platform have to be in order for her velocity to be twice as great?

(answer = four times higher)

3. A spring whose spring constant, k, is 850 N/m is compressed OAO m. What is the maximum speed it can give to a 500. g (0.5 kg) ball? (answer = 16.5 mls)

Worksheet 7 p. 1 adg, 2112/2009)

4. If the spring in #3 were compressed twice as much, how many times greater would the velocity of the ball be? (answer twice as fast)

5. A bullet with a mass of 10 g (0.010 kg) is fired from a rifle with a barrel that is 85 cm (0.85 m)

a. Assuming that the force exerted by the expanding gas to be a constant 5500 N over the length of the gun barrel, what speed would the bullet reach? (answer = 967 mls)

b. If the gun barrel is longer, how would this change the exit velocity of the bullet? (greater)

6. A 24 kg child descends a 5.0 m high slide and reaches the ground with a speed of2.8 m/s. a. How much energy was dissipated due to friction in the process? (answer 1081.92

J)

b. Do a pie chart analysis of this situation, using an accurate % of the pie to represent the amount of Ediss in the process. (1000/0 Eg goes to 80/0 Ekand 920/0 dissipated energy)

7. Remember the Wyle Coyote shot from a cannon in the Roadrunner cartoons? Suppose a scrawny 20 kg Wyle was shot straight up with an initial velocity of +50 mls. a. Assuming that his entire initial Ek was transformed into E , what is g

the maximum height he could reach? (answer = 127.55 m)

b. Suppose that 200/0 of his initial Ek were lost due to friction with the

air (air resistance). What is the maximum height he could reach? (answer 102 m)

Worksheet 7 p.2 adg, 2/] 2/2009)