Conservation of Energy

Poll

A B C

A ball rolls down each of the ramps shown. The ball is moving the fastest at the bottom of the ramp for ramp

1. A

2. B

3. C

4. None of the above; because it will have the same speed at the bottom of each ramp.

Poll

A B CA roller coaster rolls down each of the tracks shown. If friction is negligible, in which case is the roller coaster moving fastest at the bottom of the hill.1. A

2. B

3. C

4. None of the above; because it will have the same speed at the bottom of each ramp.

Conservative Force

A force for which one can write a potential energy function is called a conservative force. The change in potential energy depends on the system’s initial and final state and not on the path it takes to get there.

Spring

Another conservative force is the force on a spring to stretch or compress it a distance x from equilibrium.

Hooke’s Law:

F

Example

If you hang 0.5 kg from a spring and it stretches a distance 10 cm, what is its stiffness k?

Poll

If you hang 0.25 kg from a spring of stiffness 20 N/m, how far will the spring stretch from equilibrium?

1. 0.12 m

2. 0.012 m

3. 0.49 m

4. 0.049 m

5. 0.20 m

Poll

Suppose that a force of 2 N is needed to stretch a certain spring 0.05 m. What force is needed to stretch it 0.1 m?

1. 1 N

2. 2 N

3. 4 N

4. 8 N

5. 16 N

Elastic Potential Energy

Energy associated with the stretch or compression of a spring is called elastic potential energy.

More stretched (or more compressed), greater elastic potential energy.

Less stretched (or less compressed), less elastic potential energy.

Not stretched (or not compressed), zero elastic potential energy.

Vertical mass-on-spring

Treat the equilibrium length of the spring as if it is the unstretched length of the spring. Then, you can neglect the gravitational force on the object.

The spring becomes a compressible spring.

Example

You hang 0.25 kg on a spring of stiffness 10 N/m. You pull it down 0.05 m from its equilibrium position and release it from rest. How fast is it moving when it reaches the equilibrium position?

Example

You use a compressible spring in a dart gun to shoot a dart upward. The dart has a mass of 10 grams, the spring has a stiffness of 200 N/m, and you initially compress the spring 0.05 m. If you release it from rest, how fast is the dart moving when it leaves the spring?

Example

For the dart in the previous question, how high will it go (assuming no air resistance)?

Example

A gymnast jumps onto a springboard made of a platform and four identical parallel springs. Her mass is 50 kg and she is moving with a speed of 0.7 m/s when she hits the platform. All four springs compress a distance of 5 cm. Assuming negligible change in gravitational potential energy, what is the stiffness of the springs?

Example

You build a model of a bungee jumper using a spring of stiffness 5 N/m and a mass of 0.2 kg. The fixed end of the spring is at y=0, and the spring is 10 cm long. You release the object at y=0. How far does the object fall before being “jerked” upward?

Power

Power is the rate at which energy is transferred.

SI unit of power is the watt (W). 1 W = 1 J/s

Poll

How much energy does a 100-W light bulb use in 1 minute?

1. 100 J

2. 60 J

3. 600 J

4. 6000 J

5. none of the above

Poll

While exercising, I transfer 250 calories of energy to the exercise machine in 20 minutes. What is the power in units of cal/min?

1. 250 cal/min

2. 125 cal/min

3. 12.5 cal/min

4. 1.25 cal/min

5. 5000 cal/min

Example

If “burning” 12.5 cal/min while exercising, what is your power output in watts?

Example

The HP electric company charges about $0.095102 per kW hour. How much does it cost to operate a 100 W light bulb for 24 hours?

Example

When exercising, I do intervals. Suppose that for one minute, I burn 5 cal/min, then in the next minute 10 cal/min, then in the next minute 15 cal/min, and for the next minute 20 cal/min. How much energy do I burn in four minutes?

Example

Energy can be transferred via radiation. For sunlight incident on a given area, the energy incident on the area is 1353 J/m2/s or 1353 W/m2. A typical solar cell efficiency in converting radiant energy to electrical energy is 20%. How much electrical energy can one get from a 1m2 area solar panel during 8 hours of sunlight?