Interactive Lecture Demonstration

Introduction to Conservation of Energy

Studio Physics I

Energy of a Cart on an Inclined Plane (with very low friction)

In this activity, we consider a cart given a quick push up an incline. The cart rolls up the ramp moving away from the motion detector, slowing down as it goes, reaches its highest point and then rolls back down the ramp speeding up on the way. The cart is then caught at the bottom of the track. This motion was demonstrated by your professor.

1. Sketch predictions of the acceleration-time, velocity-time and position-time graphs for the cart (with almost no friction). Show the push and the catch.

a) As the cart moves up the inclined plane, the direction of the net force on the cart is: (choose one from each line)

Up the incline Down the incline Zero

And

Increasing in strength Decreasing in strength Constant

b) At the cart's highest point, the direction of the net force on the cart is :

Up the incline Down the incline Zero

And

Increasing in strength Decreasing in strength Constant

c) As the cart moves back down the inclined plane, the direction of the net force on the cart is:

Up the incline Down the incline Zero

And

Increasing in strength Decreasing in strength Constant

3. Open a file called "conserv1.mbl". You can get the file under the “activities” icon on the web page or from the CD. A graph of the cart's position as a function of time is shown. Compare this graph to the prediction that you made in Step 1 above and sketch the correct graph now. Change the graph to display the velocity of the cart as a function of time. (This is done by placing the cursor tip over the word "position" on the y-axis of the graph and clicking the left mouse button. Uncheck “position” and check “velocity”.) Compare this graph to the prediction that you made in Step 1 above and sketch the correct graph now. Do the same for the acceleration of the cart as a function of time graph.

4. Sketch a prediction graph of the kinetic energy (the energy due to motion) of the cart over time as it moves. Keep in mind that the kinetic energy (KE) is 1/2mv2where m is the mass of the cart (in this case 1 kg) and v is the velocity. When is the kinetic energy zero? When is it a maximum? (Write your answers to the right of your sketch).

5. Open a file called "conserv2.mbl" that is available on the web page or CD. The graph displayed is a graph of the cart's velocity as a function of time. Change the graph so that it displays the cart’s kinetic energy (see step 3 above if you don't remember how to do this). Compare the actual graph of the cart's kinetic energy to your prediction, and sketch the correct graph now. Mark on your sketch the regions representing the push and catch.

6. Sketch a prediction graph of the potential energy of the cart (the energy due to raising the cart's mass in the gravitational field of the earth). Define the potential energy to be zero at the height at which the cart is first pushed. When is the potential energy zero? When is it a maximum? (Write your answers to the right of the graph)

7. Open the file called "conserv3.mbl" in your class10 activity folder. The graph displayed is a graph of the cart's position as a function of time. Change the graph so that it displays the cart gravitational potential energy (see step 3 above if you don't remember how to do this). Compare the actual graph of the cart's gravitational potential energy to your prediction, and sketch the correct graph now.

8. Sketch below your prediction of the total mechanical energy (the sum of the kinetic and potential energies) of the cart over time as it moves. Describe in words (using complete sentences) the nature of the mechanical energy of the cart after the push and before the catch ( is it increasing, decreasing….?).

9. Open the file called "conserv4.mbl" that is available on the web page or CD. Graphs of the cart's gravitational potential energy and kinetic energy are displayed. The bottom graph shows the cart mechanical energy. Compare the actual graph of the cart's mechanical energy to your prediction and sketch the correct graph now. Mark on your sketch the push and catch.

10. State in complete sentences the main differences in the graphs of the cart’s KE, Ug, and mechanical energy.

11. Explain what "conserved" means. Is the kinetic energy of the cart conserved? How do you know this? Is the gravitational potential energy of the cart conserved? How do you know this? Is the mechanical energy conserved? How do you know this?

12. Where does the cart get the initial energy? Where does the energy go at the end of the motion?

13. Consider a cart that is released from rest at the top of an incline and is allowed to roll to the bottom of the incline with very little friction present. When the cart reaches the bottom of the incline, it has a speed of 10 m/s.

a) If the incline is made to be just as high but much longer, will the cart's speed at the bottom of the incline be greater than 10 m/s? Less than 10 m/s? or 10 m/s? Explain why you chose the answer that you did (using complete sentences).

b) If the incline is made to be just as high but much shorter, will the cart's speed at the bottom of the incline be greater than 10 m/s? Less than 10 m/s? or 10 m/s? Explain why you chose the answer that you did (using complete sentences).

c) If the incline is made to be just as long but much higher, will the cart's speed at the bottom of the incline be greater than 10 m/s? Less than 10 m/s? or 10 m/s? Explain why you chose the answer that you did (using complete sentences).

14. We often need to know the angle that an incline makes with a horizontal surface. Describe at least two ways that one could determine the angle () that the track makes with the horizontal table top. How can we find sin  without using a calculator or protractor?

15. We used the motion detector to measure the position of the cart on the track. The location of the motion detector is chosen to be our origin. Can we use the data collected by the motion sensor and our knowledge of the track's angle of inclination () to calculate the height of the cart above the tabletop? If so, how? Can we use the data collected by the motion sensor and our knowledge of the track's angle of inclination () to calculate the height of the cart above the position of the motion sensor? If so, how?

16. PE=mgh where h is the height above the starting point, g is the acceleration due to gravity, and m is the mass of the cart. By using trigonometry we can calculate h from x, which is the distance of the cart from the motion detector. Derive the expression for h in terms of x, xi(the initial position) and ø (the angle the track makes with the horizontal)?

17. Suppose our cart starts out 40 cm (measured along the track) from the motion detector. The length of the track is 2meters, and the height of the raised end of the track is 20 cm. For this situation:

sin  = ______= ______

hi (cart's starting height above the motion detector) = ______

18. How would your graphs of Ug, KE and mechanical energy change if we chose a different point (one other than the cart's initial height) to call our "zero height"?