TUTORIAL 1 SIMPLE HARMONIC MOTION Instructor: Kazui Tolich
About tutorials 2 Tutorials are conceptual exercises that should be worked on in groups. Each slide will consist of a series of questions that you should discuss with the students sitting around you. There will be a few clicker questions per session. Clicker questions are shown in red. There will be several teaching assistants wandering around the roo to assist you. If you are having difficulty, they will ask you leading questions to help you understand the idea.
I. Hook s law for spring 3 A block of ass on a frictionless surface is attached to an ideal spring, as shown in figure 1. The spring, with a spring constant k, is fixed to the wall. The dashed line indicates the position of the right edge of the spring when the spring is relaxed, and the block is at its equilibriu position. Figures 2 and 3 show the block held in place a distance A to the right and left of the equilibriu position, respectively. Figure 1 A Figure 2 A Figure 3
I. Hook s law for spring 4 A. In the boxes at right, draw arrows to represent the directions of: Figure 1 The position of the block, x, taking x = 0 when the block is at the equilibriu position. The force on the block by the spring, F. A Figure 2 Use the arrows that you drew to explain why the inus sign is necessary in Hook s law: F = kx. A Figure 3 Arrows for figure 2 Position of block Force on block by spring Arrows for figure 3 Position of block Force on block by spring B. What is the net force on the block when it is held in place as shown in figure 2?
I. Hook s law for spring 5 Figure 1 A Figure 2 A Figure 3 Suppose the hand in figure 2 were suddenly reoved. After the hand is reoved, how would the force on the block by the spring be related to the net force on the block? C. Use Newton s second law to write an expression for the acceleration, a, of the block in ters of x, k, and for an instant after the block has been released.
II. Siple haronic otion 6 A pendulu with a ass is hung directly above the blockspring syste fro section I. The length of the pendulu is l. The paraeters of the systes have been chosen such that the block is always directly below the pendulu bob. A. Draw a vector to represent the net force on the block and the bob when it is on the far left as shown. Net force on block Net force on bob k Equilibriu position l
II. Siple haronic otion 7 Suppose that the asses of the block and the pendulu bob were doubled, and the block and bob were then released fro rest at the far left position. The following questions serve as a guide to help you deterine whether the block would reain directly below the pendulu bob at all ties. 1. As a result of doubling the asses of the block and pendulu bob, would the following quantities increase by a factor of 2, decrease by a factor of 2, or reain the sae? the agnitude of the net force on the block, F,-., 12345, or the bob, F,-., 131, when it is at the far left n F,-., 12345 n F,-., 131
II. Siple haronic otion 8 1. As a result of doubling the asses of the block and pendulu bob, would the agnitude of the acceleration of each object, a 12345 or a 131, when it is at the far left increase by a factor of 2, decrease by a factor of 2, or reain the sae? n a 12345 n a 131 would the tie it takes for each object to travel fro the far left position to the equilibriu position, t 12345 or t 131, increase, decrease, or reain the sae? n t 12345 n t 131
II. Siple haronic otion 9 2. Check that your answers regarding t 12345 and t 131 are consistent with the relationships below: (Hint: For each object, what is the relationship between t and the period of oscillation, T?) T 12345 = 2π k (period of oscillation for a ass on a spring) T 131 = 2π l g (period of oscillation for a siple pendulu) 3. Will the block and pendulu bob still ove together after their asses have been doubled? If not, describe what additional changes could be ade so that the block and pendulu bob would again ove together.
III. Energy of a siple haronic oscillator 10 The diagra at right is a plot of the total energy of a horizontal block-spring syste as a function of the position of the block with respect to its equilibriu position. The block oscillates with a axiu distance fro equilibriu of A. E E tot -A +A x
III. Energy of a siple haronic oscillator 11 A. What feature of the diagra shows that the total energy of the syste is conserved as the block oscillates? B. Deterine what fraction of the total energy is potential energy, U = 1 2 kx B, when the block is at x = +A and x = + A 2. Plot the potential energy stored in the spring as a function of x. E E tot -A +A x
III. Energy of a siple haronic oscillator 12 C. On the sae axes, plot the kinetic energy of the block, K, as a function of x. (Hint: What function ust be added to the potential energy to equal the total energy of the syste?) Label your graphs so that you can easily distinguish the kinetic energy K fro the potential energy U. E E tot D. Is the tie that the block takes to ove fro x = 0 to x = + A 2 greater than, less than, or equal to the tie that the block takes to ove fro x = + A 2 to x = +A? Explain how your answer is consistent with your graph of the kinetic energy. -A +A x
III. Energy of a siple haronic oscillator 13 E. Consider two points, P and Q, to the right of the block s equilibriu position. Point P has position x = + A 2. Point Q is the point for which the kinetic energy and the potential energy of the syste are each equal to half the total energy. 1. Is point Q to the left of, to the right of, or at the sae position as point P? Mark the locations of points P and Q on the x-axis above. E E tot 2. When the block is at point P, is the kinetic energy of the syste greater than, less than, or equal to the potential energy? Explain how you can tell fro the graph. 3. Calculate the ratio of the kinetic energy to the potential energy when the block is at point P (i.e., at x = + A 2). -A +A x
III. Energy of a siple haronic oscillator 14 F. Suppose that the block were replaced by a block with half the ass and released fro rest at x = +A. 1. Describe any resulting changes in the three energy graphs (E.3.G2, U, K). Explain. E E tot 2. Is it possible to deterine the period of oscillation of a ass-spring syste using inforation fro energy graphs alone? If so, describe the steps you would take to deterine the period. If not, state what other inforation you would need. -A +A x
III. Energy of a siple haronic oscillator 15 G. Suppose instead that the original block were released fro rest at x = A 2 and oved between the positions x = A 2 and x = + A 2. 1. As a result of this change, would the total energy increase, decrease, or reain the sae? Explain. E E tot, old 2. On the axes at right, graph the total energy, E.3.G2,,-H, potential energy U,-H, and kinetic energy K,-H for the new otion. -A +A x
III. Energy of a siple haronic oscillator 16 3. Does any part of the new set of graphs exactly coincide with any part of the previous set of graphs? Explain. E E tot, old 4. Is the tie that the block takes to ove fro x = 0 to x = + A 2 greater than, less than, or equal to the tie that the block took to ove fro x = 0 to x = + A 2 before the aplitude of oscillation was reduced? Explain. -A +A x