Slide 1 / 102 Slide 2 / 102 AP Physics C - Mechanics Simple Harmonic Motion 2015-12-05 www.njctl.org Slide 3 / 102 Table of Contents Click on the topic to go to that section Spring and a Block · Energy of SHM · SHM and UCM · Simple and Physical Pendulums · Sinusoidal Nature of SHM ·
Slide 4 / 102 Spring and a Block Return to Table of Contents Slide 5 / 102 Periodic Motion Periodic motion describes objects that oscillate about an equilibrium point. This can be a slow oscillation - like the earth orbiting the sun, returning to its starting place once a year. Or very rapid oscillations such as alternating current or electric and magnetic fields. Simple harmonic motion is a periodic motion where there is a force that acts to restore an object to its equilibrium point - it acts opposite the force that moved the object away from equilibrium. The magnitude of this force is proportional to the displacement of the object from the equilibrium point. Slide 6 / 102 Simple Harmonic Motion Simple harmonic motion is described by Hooke's Law. Robert Hooke was a brilliant scientist who helped survey and architect London after the Great Fire of London in 1666, built telescopes, vaccums, observed the planets, used microscopes to study cells (the name cell comes from Hooke's observations of plant cells) and proposed the inverse square law for gravitational force and how this force explained the orbits of planets. Unfortunately for Robert Hooke, he was a contemporary of Sir Isaac Newton and the two men were not friends. In fact, there 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 Hooke's Law Hooke developed his law to explain the force that acts on an elastic spring that is extended from its equilibrium (rest position - where it is neither stretched nor compressed). If the spring is stretched in the positive x direction, a restorative force will act to bring it back to its equilibrium point - a negative force: k is the spring constant and its units are N/m. Slide 8 / 102 Hooke's Law For an object to be in simple harmonic motion, the force has to be linearly dependent on the displacement. If it is proportional to the square or any other power of the displacement, then the object is not in simple harmonic motion. The force is not constant, so the acceleration is not constant either. This means the kinematics equations cannot be used to solve for the velocity or position of the object. Slide 9 / 102 1 A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring?
Slide 9 (Answer) / 102 1 A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring? Answer [This object is a pull tab] Slide 10 / 102 2 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring system. How far is the mass from equilibrium? Slide 10 (Answer) / 102 2 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring system. How far is the mass from equilibrium? Answer [This object is a pull tab]
Slide 11 / 102 3 A spring exerts a force of 50N on the mass in a mass-spring system when it is 2.0m from equilibrium. What is the spring's spring constant? Slide 11 (Answer) / 102 3 A spring exerts a force of 50N on the mass in a mass-spring system when it is 2.0m from equilibrium. What is the spring's spring constant? Answer [This object is a pull tab] Slide 12 / 102 Simple Harmonic Motion 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) The minimum force exerted on the mass is when the spring is not stretched at all (x = 0) F = 0 (when x = 0) x -A 0 A
Slide 13 / 102 Simple Harmonic Motion When the spring is all the way compressed: x -A 0 A The displacement is at the negative amplitude. · The force of the spring is in the positive direction. · The acceleration is in the positive direction. · The velocity is zero. · Slide 14 / 102 Simple Harmonic Motion When the spring is at equilibrium and heading in the positive direction: x -A 0 A The displacement is zero. · The force of the spring is zero. · The acceleration is zero. · The velocity is positive and at a maximum. · Slide 15 / 102 Simple Harmonic Motion When the spring is all the way stretched in the positive direction: x -A 0 A The displacement is at the positive amplitude. · The force of the spring is in the negative direction. · The acceleration is in the negative direction. · The velocity is zero. ·
Slide 16 / 102 Simple Harmonic Motion When the spring is at equilibrium and heading in the negative direction: x -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 / 102 4 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum? A x = A B x = 0 C x = -A D x = A and x = -A E All of the above Slide 17 (Answer) / 102 4 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum? A x = A B x = 0 C x = -A D x = A and x = -A E All of the above Answer D [This object is a pull tab]
Slide 18 / 102 5 At which location(s) is the magnitude of the force on the mass in a mass-spring system a minimum? A x = A B x = 0 C x = -A D x = A and x = -A E All of the above Slide 18 (Answer) / 102 5 At which location(s) is the magnitude of the force on the mass in a mass-spring system a minimum? A x = A B x = 0 C x = -A D x = A and x = -A Answer E All of the above B [This object is a pull tab] Slide 19 / 102 Vertical Mass-Spring System If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational y = y 0 force. The displacement is now y = 0 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 Vertical Mass-Spring System 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 / 102 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? Slide 21 (Answer) / 102 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? Answer [This object is a pull tab]
Slide 22 / 102 Springs in Parallel 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? Slide 23 / 102 Springs in Parallel For a given applied force, mg, the new springs will stretch only half as much as the original spring. Let y equal the distance the springs stretch when the mass is attached. m m The spring constant of each piece is m twice the spring constant of the original spring. Slide 24 / 102 Springs in Parallel Next attach just one mass to the two spring combination. Let's calculate the effective spring constant of two springs in parallel, each with spring constant = k', by using a free body diagram. y is the distance each spring is stretched. ky ky m By cutting a spring in half, and then attaching each piece to a mass, the effective mg spring constant is quadrupled. The spring system is four times as stiff as the original spring.
Slide 25 / 102 Springs in Parallel For identical springs in parallel, the effective spring constant is just twice the spring constant of either spring. We cannot generally apply this to springs with different spring constants.. Why? m Slide 26 / 102 Springs in Parallel If the springs had different spring constants, then one spring would be stretched more than the other - and the mass would 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. m Slide 27 / 102 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 force, each spring stretches a distance y 1 and y 2 where the total stretch of the two springs is y T . F is given and constant y 1 y T = y 1 + y 2 y 2 m
Slide 28 / 102 Springs in Series y 1 The effective spring constant of the two springs in series is: y T = y 1 + y 2 y 2 k eff is less than either one of the m spring constants that were joined together. The combination is less stiff then either spring alone with the mass. Slide 29 / 102 Energy of SHM Return to Table of Contents 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 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.
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