Slide 1 / 102 Slide 2 / 102 AP Physics C - Mechanics Simple Harmonic Motion 2015-12-05 www.njctl.org Slide 3 / 102 Slide 4 / 102 Table of Contents Click on the topic to go to that section Spring and a Block · Energy of SHM · SHM and UCM · Spring and a Block Simple and Physical Pendulums · Sinusoidal Nature of SHM · Return to Table of Contents Slide 5 / 102 Slide 6 / 102 Periodic Motion Simple Harmonic Motion Periodic motion describes objects that oscillate about an Simple harmonic motion is described by Hooke's Law. equilibrium point. This can be a slow oscillation - like the earth orbiting the sun, returning to its starting place once a year. Or Robert Hooke was a brilliant scientist who helped survey and very rapid oscillations such as alternating current or electric architect London after the Great Fire of London in 1666, built and magnetic fields. telescopes, vaccums, observed the planets, used microscopes to study cells (the name cell comes from Hooke's observations Simple harmonic motion is a periodic motion where there is of plant cells) and proposed the inverse square law for a force that acts to restore an object to its equilibrium point - it gravitational force and how this force explained the orbits of acts opposite the force that moved the object away from planets. equilibrium. Unfortunately for Robert Hooke, he was a contemporary of Sir The magnitude of this force is proportional to the displacement Isaac Newton and the two men were not friends. In fact, there of the object from the equilibrium point. are no pictures of Hooke - possibly due to Newton's influence - and Newton gave no credit to Hooke for any of his physics work.
Slide 7 / 102 Slide 8 / 102 Hooke's Law Hooke's Law Hooke developed his law to explain the force that acts on an For an object to be in simple harmonic motion, the force has to elastic spring that is extended from its equilibrium (rest be linearly dependent on the displacement. If it is proportional position - where it is neither stretched nor compressed). If the to the square or any other power of the displacement, then the spring is stretched in the positive x direction, a restorative object is not in simple harmonic motion. force will act to bring it back to its equilibrium point - a negative force: The force is not constant, so the acceleration is not constant either. k is the spring constant and its units are N/m. This means the kinematics equations cannot be used to solve for the velocity or position of the object. Slide 9 / 102 Slide 9 (Answer) / 102 1 A spring whose spring constant is 20N/m is 1 A spring whose spring constant is 20N/m is stretched stretched 0.20m from equilibrium; what is the magnitude of the 0.20m from equilibrium; what is the magnitude of the force exerted by the spring? force exerted by the spring? Answer [This object is a pull tab] Slide 10 / 102 Slide 10 (Answer) / 102 2 A spring whose spring constant is 150 N/m 2 A spring whose spring constant is 150 N/m exerts a force exerts a force of 30N on the mass in a mass-spring system. How far is of 30N on the mass in a mass-spring system. How far is the mass from equilibrium? the mass from equilibrium? Answer [This object is a pull tab]
Slide 11 / 102 Slide 11 (Answer) / 102 3 A spring exerts a force of 50N on the mass in a 3 A spring exerts a force of 50N on the mass in a mass-spring system when it is 2.0m from mass-spring system when it is 2.0m from equilibrium. What is the spring's spring constant? equilibrium. What is the spring's spring constant? Answer [This object is a pull tab] Slide 12 / 102 Slide 13 / 102 Simple Harmonic Motion Simple Harmonic Motion When the spring is all the way compressed: The maximum force exerted on the mass is when the spring is most stretched or compressed (x = -A or +A): F = -kA (when x = -A or +A) x The minimum force exerted on the mass is when the spring -A 0 A is not stretched at all (x = 0) F = 0 (when x = 0) The displacement is at the negative amplitude. · The force of the spring is in the positive direction. · x The acceleration is in the positive direction. · -A 0 A The velocity is zero. · Slide 14 / 102 Slide 15 / 102 Simple Harmonic Motion Simple Harmonic Motion When the spring is at equilibrium and When the spring is all the way stretched in the positive heading in the positive direction: direction: x x -A 0 A -A 0 A The displacement is zero. · The displacement is at the positive amplitude. · The force of the spring is zero. · The force of the spring is in the negative direction. · The acceleration is zero. · The acceleration is in the negative direction. · The velocity is positive and at a maximum. · The velocity is zero. ·
Slide 16 / 102 Slide 17 / 102 Simple Harmonic Motion 4 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum? When the spring is at equilibrium and heading A x = A in the negative direction: B x = 0 C x = -A D x = A and x = -A x E All of the above -A 0 A The displacement is zero. · The force of the spring is zero. · The acceleration is zero. · The velocity is negative and at a maximum. · Slide 17 (Answer) / 102 Slide 18 / 102 5 At which location(s) is the magnitude of the force 4 At which location(s) is the magnitude of the force on the mass in a mass-spring system a minimum? on the mass in a mass-spring system a maximum? A x = A A x = A B x = 0 B x = 0 C x = -A C x = -A D x = A and x = -A D x = A and x = -A E All of the above E All of the above Answer D [This object is a pull tab] Slide 18 (Answer) / 102 Slide 19 / 102 5 At which location(s) is the magnitude of the force Vertical Mass-Spring System on the mass in a mass-spring system a minimum? A x = A If the spring is hung B x = 0 vertically, the only change is in the C x = -A equilibrium position, D x = A and x = -A which is at the point Answer where the spring force E All of the above B equals the gravitational y = y 0 force. The displacement is now y = 0 [This object is a pull tab] measured from the new equilibrium position, y = 0. The value of k for an unknown spring can be found via this arrangement.
Slide 20 / 102 Slide 21 / 102 6 An object of mass 0.45 kg is attached to a spring with Vertical Mass-Spring System k = 100 N/m and is allowed to fall. What is the maximum distance that the mass reaches before it stops and begins heading back up? Use Newton's Second Law in the y direction when the mass is at rest at its new ky 0 equilibrium position. mg y = y 0 y = 0 Slide 21 (Answer) / 102 Slide 22 / 102 Springs in Parallel 6 An object of mass 0.45 kg is attached to a spring with k = 100 N/m and is allowed to fall. What is the maximum distance that the mass reaches before it stops and begins heading back up? Take a spring with spring constant k, and cut it in half. What is the spring constant, k' of each of the two new springs? Answer [This object is a pull tab] Slide 23 / 102 Slide 24 / 102 Springs in Parallel Springs in Parallel Next attach just one mass to the two spring combination. Let's For a given applied force, mg, the new springs will stretch calculate the effective spring constant of two springs in parallel, only half as much as the original spring. Let y equal the each with spring constant = k', by using a free body diagram. distance the springs stretch when the mass is attached. y is the distance each spring is stretched. ky ky m m m By cutting a spring in half, and then attaching each piece to a mass, the effective mg The spring constant of each piece is spring constant is quadrupled. The spring m twice the spring constant of the system is four times as stiff as the original original spring. spring.
Slide 25 / 102 Slide 26 / 102 Springs in Parallel Springs in Parallel For identical springs in parallel, the effective spring constant is If the springs had different spring constants, then one spring just twice the spring constant of either spring. We cannot would be stretched more than the other - and the mass would generally apply this to springs with different spring constants.. feel a net torque and rotate. It would be hard to predict what the behavior of the mass would be. So, the problems will be limited to identical springs in parallel. Why? m m Slide 27 / 102 Slide 28 / 102 Springs in Series Springs in Series We don't have this limitation for springs in series, as they contact the mass at only one point. Take two springs of spring constants k 1 and k 2 , and attach them to each other. For a given y 1 force, each spring stretches a distance y 1 and y 2 where the total stretch of the two springs is y T . The effective spring constant of the two springs in series is: y T = y 1 + y 2 F is given and constant y 1 y 2 y T = y 1 + y 2 k eff is less than either one of the m spring constants that were joined together. The combination is less y 2 stiff then either spring alone with the mass. m Slide 29 / 102 Slide 30 / 102 Energy of SHM The spring force is a conservative force which allows us to calculate a potential energy associated with simple harmonic motion. The force is not constant, so in addition to not being able to use the kinematics equations to predict motion, the potential energy Energy of SHM can't be found by taking the negative of the work done by the spring on the block where work is found by multiplying a constant force by the displacement. Return to Table of Contents
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