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Transcript
LECTURE 3 PENDULUM AND ENERGY Instructor: Kazumi Tolich Lecture 3 2 ¨ Reading chapter 13.5 – 13.6 Mechanical energy conservation in oscillatory motion ¤ Pendulum ¤ Simple pendulum n Physical pendulum n Mechanical energy conservation 3 ¨ ¨ If non-conservative force (friction, air resistance etc) is not doing any work, the mechanical energy is conserved. The total mechanical energy of a mass on a spring is 𝐸 = 𝐾 + 𝑈 = &'𝑚𝑣 * + &'𝑘𝑥 * = &'𝑘𝐴* . Quiz: 1 ¨ ¨ When does a body moving in simple harmonic motion have the maximum acceleration? Choose all that apply. When it has A. B. C. D. E. F. G. maximum velocity. zero velocity. maximum kinetic energy. minimum kinetic energy. maximum potential energy. minimum potential energy. zero displacement. Quiz: 3-1 answer ¨ When it has A. B. C. D. E. F. G. ¨ maximum velocity. zero velocity. maximum kinetic energy. minimum kinetic energy. maximum potential energy. minimum potential energy. zero displacement. When the spring is maximally stretched/compressed, the force on the mass is maximum resulting in the maximum acceleration. Quiz: 2 6 ¨ A mass oscillates in simple harmonic motion with amplitude 𝐴. If the mass is doubled, but the amplitude is not changed, what will happen to the total mechanical energy of the mass-spring system? A. B. C. Increases Stays the same Decreases Quiz: 3-2 answer 7 ¨ ¨ ¨ Stays the same The total mechanical energy is equal to the initial value of the elastic potential energy, which is 𝑈 = &'𝑘𝐴* . This does not depend on mass, so a change in mass will not affect the energy of the system. Example: 1 (Walker Ch. 13-53) 8 ¨ A block with a mass of 𝑚 = 0.505 kg slides on a frictionless horizontal surface with a speed of 𝑣 = 1.18 m/s. The block encounters an unstretched spring and compresses it 𝐴 = 23.2 cm before coming to rest. A. B. C. What is the force constant of this spring? For what length of time is the block in contact with the spring before it comes to rest? If the force constant of the spring is increased, does the time required to stop the block increase, decrease, or stay the same? Simple pendulum 9 ¨ ¨ ¨ The weight of the bob provides the restoring force. Simple pendula do not exhibit true simple harmonic motion for any angle. However, if the angle of oscillation 𝜃 is small, the motion is close to and can be modeled as simple harmonic motion. The period of a simple pendulum with a length 𝐿, oscillating with a small amplitude is given by 𝑇 = 2𝜋 𝐿 𝑔 Demo 1 10 ¨ Simple Pendula with Different Lengths and Masses ¤ Demonstration of the relationship between 𝐿 and 𝑇 : 𝑇 = 2𝜋 6 7 Quiz: 3 11 ¨ The graph shows the square of the period versus the length of a simple pendulum on a certain planet. What is the acceleration due to gravity on that planet in m/s2? Quiz: 3-3 answer 12 ¨ 4 m/s2 ¨ 𝑇 = 2𝜋 6 7 ¨ 𝑔 = 2𝜋 * 6 8' = 2𝜋 * 9 : 9; <' = 4 m⁄s * Quiz: 4 13 ¨ Two pendula have identical periods. One has a slightly larger amplitude than the other, but both swing through small angles compared to vertical. Which of the following must be true of the pendulum that has the larger amplitude? Choose all the apply. A. B. C. D. E. F. G. H. It has more mass than the other one. It has less mass than the other one. It is longer than the other one. It is shorter than the other one. It has slightly more energy than the other one. It has slightly less energy than the other one. It moves faster at the lowest point in its swing than the other one. It moves slower at the lowest point in its swing than the other one. Quiz: 3-4 answer 14 ¨ It moves faster at the lowest point in its swing than the other one. ¨ 𝑇 = 2𝜋 ¨ ¨ ¨ ¨ 6 7 The period of the pendulum at a location depends only on the length, independent of its amplitude or mass. The two pendula must be of the same length. The greater the amplitude, the greater distance that the bob must travel. The energy of the pendulum depends on the mass of the bob, but we have no information about it. Physical pendulum 15 ¨ ¨ A physical pendulum is a rigid object free to rotate about a horizontal axis that is not through its center of mass that oscillates when displaced from equilibrium. Physical pendula do not exhibit true simple harmonic motion for any angle. However, if the angle of oscillation is small, the motion is close to and can be modeled as simple harmonic motion with its period given by 𝑇 = 2𝜋 𝑙 𝑔 𝐼 𝑚𝑙 * where 𝑙 is the distance between the axis and the center of mass, 𝐼 is the moment of inertia about the axis, and 𝑚 is the mass of the object. Demo: 2 16 ¨ Physical pendulum ¤ Demonstration C of the relationship between 𝑇 and D : 𝑇 = 2𝜋 D 7 C ED ' Quiz: 5 17 ¨ Recall that period of oscillation of a physical pendulum is given by 𝑇 = 2𝜋 D 7 C ED ' . The moment of inertia is always proportional to the mass of the object 𝐼 ∝ 𝑚. Suppose that each of two objects is suspended from a pivot point such that the distances between the center of mass and the pivot point, 𝑙, is the same. Does it mean then that the periods of oscillation must be the same? A. B. Yes No Quiz: 3-5 answer 18 ¨ ¨ ¨ ¨ No Let 𝐼 = 𝐶𝑚 Even if 𝑙’s are the same, if the objects are of the different shapes such that 𝐶’s are different, they would have different oscillation periods. 𝑇 = 2𝜋 D 7 C ED ' = 2𝜋 C 7ED = 2𝜋 HE 7ED = 2𝜋 H 7D Example: 2 19 ¨ Consider the two pendula shown. In Case 1 a stick of mass 𝑀 is pivoted at one end and used as a pendulum. In Case 2 a point particle of mass 𝑀 is attached to the center of the same stick. Calculate the periods of oscillation. The moment J of inertia of a rod about its end is 𝐼 = 𝑀𝐿* , K where 𝐿 is the length of the rod, and the moment of inertia of a point mass is 𝐼 = 𝑀𝑟 * , where 𝑟 is the distance between the mass and the pivot.