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Similar to simple harmonic motion
simple harmonic motion
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- 2. Simple Harmonic Motion simple harmonic motion (SHM) – vibration about an equilibrium position in which a restoring force is proportional to the displacement from equilibrium two common types of SHM are a vibrating spring and an oscillating pendulum springs can vibrate horizontally (on a frictionless surface) or vertically
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- 4. SHM and OscillatingSprings in an oscillating spring, maximum velocity (with Felastic = 0) is experienced at the equilibrium point; as the spring moves away from the equilibrium point, the spring begins to exert a force that causes the velocity to decrease the force exerted is maximum when the spring is at maximum displacement (either compressed or stretched)
- 5. SHM and OscillatingSprings at maximum displacement, the velocity is zero; since the spring is either stretched or compressed at this point, a force is again exerted to start the motion over again in an ideal system, the mass-spring system would oscillate indefinitely
- 6. SHM and OscillatingSprings damping occurs when friction slows the motion of the vibrating mass, which causes the system to come to rest after a period of time if we observe a mass-spring system over a short period of time, damping is minimal and we can assume an ideal mass-spring system
- 7. SHM and OscillatingSprings in a mass-spring system, the spring force is always trying to pull or push the mass back toward equilibrium; because of this, we call this force a restoring force in SHM, the restoring force is proportional to the mass’ displacement; this results in all SHM to be a simple back-and-forth motion over the same path
- 8. Hooke’s Law in1678, Robert Hooke proposed this simple relationship between force and displacement; Hooke’s Law is described as: Felastic = -kx where Felastic is the spring force, k is the spring constant x is the maximum displacement from equilibrium
- 9. Hooke’s Law thenegative sign shows us that the force is a restoring force, always moving the object back to its equilibrium position the spring constant has units of Newtons/meter the spring constant tells us how resistant a spring is to being compressed or stretched (how many Newtons of force are required to stretch or compress the spring 1 meter) when stretched or compressed, a spring has potential energy
- 10. Simple Pendulum simplependulum – consists of a mass (called a bob) that is attached to a fixed string; we assume that the mass of the bob is concentrated at a point at the center of mass of the bob and the mass of the string is negligible; we also disregard friction and air resistance
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- 12. Simple Pendulum forsmall amplitude angles (less than 15°), a pendulum exhibits SHM at maximum displacement from equilibrium, a pendulum bob has maximum potential energy; at equilibrium, this PE has been converted to KE amplitude – the displacement from equilibrium
- 13. Period and Frequency period (T) – the time, in seconds, to execute one complete cycle of motion; units are seconds per 1 cycle frequency (f) – the number of complete cycles of motion that occur in one second; units are cycles per 1 second (also called hertz)
- 14. Period and Frequency frequency is the reciprocal of period, so the period of a simple pendulum depends on the length of the string and the value for free-fall acceleration (in most cases, gravity)
- 15. Period of aSimple Pendulum notice that only length of the string and the value for free-fall acceleration affect the period of the pendulum; period is independent of the mass of the bob or the amplitude
- 16. Period of aMass-Spring System period of a mass-spring system depends on mass and the spring constant notice that only the mass and the spring constant affect the period of a spring; period is independent of amplitude (only for springs that obey Hooke’s Law)
- 17. Comparison of aPendulum and an Oscillating Spring