Unit 2: Simple Harmonic Motion (SHM) THE MOST COMMON FORM OF MOTION FALL 2015
Objectives: Define SHM specifically and give an example. Write and apply formulas for finding the frequency f, period T, w angular frequency, velocity v, or acceleration a in terms of displacement x or time t. Describe the motion of pendulums and spring systems and calculate the length required to produce a given frequency. Write and apply Hooke s Law for objects moving with simple harmonic motion.
Displacement in SHM x m x = -A x = 0 x = +A Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left. The maximum displacement is called the amplitude A.
Periodic Motion Periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. f 1 T Period, T, is the time for one complete oscillation. (seconds,s) Amplitude A Frequency, f, is the number of complete oscillations per second. Hertz (s -1 )
Example 1: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? T 15 s 30 cylces 0.50 s x F Period: T = 0.500 s f 1 1 T 0.500 s Frequency: f = 2.00 Hz
Simple Harmonic Motion, SHM Simple harmonic motion: periodic motion in the absence of friction (Amplitude is always the same/ Energy is Conserved) produced by a restoring Force that is directly proportional to the displacement and oppositely directed (restoring to equilibrium). Displacement does not affect the frequency/ The energy of the system does not affect the frequency.
7 https://www.youtube.com/watch? v=vngkomoukgi https://www.youtube.com/watch?v=sz 541Luq4nE
https://www.youtube.com /watch?v=eeyrkw8v7vg
Conceptual Example 4. Moving Lights 10 Over the entrance to a restaurant is mounted a strip of equally spaced light bulbs, as Figure 10.13a illustrates. Starting at the left end, each bulb turns on in sequence for one-half second. Thus, a lighted bulb appears to move from left to right. Once the apparent motion of a lighted bulb reaches the right side of the sign, the motion reverses. The lighted bulb then appears to move to the left, as part b of the drawing indicates. Thus, the lighted bulb appears to oscillate back and forth. Is the apparent motion simple harmonic motion? No. Speed is constant.
Damped Harmonic Motion In the presence of energy dissipation, the amplitude of oscillation decreases as time passes, and the motion is no longer simple harmonic motion. Instead, it is referred to as damped harmonic motion, the decrease in amplitude being called damping. 13
14 The smallest degree of damping that completely eliminates the oscillations is termed critical damping, and the motion is said to be critically damped. When the damping exceeds the critical value, the motion is said to be overdamped. In contrast, when the damping is less than the critical level, the motion is said to be underdamped (curves 2 and 3).
16 The maximum excursion from equilibrium is the amplitude A of the motion. The shape of this graph is characteristic of simple harmonic motion and is called sinusoidal, because it has the shape of a trigonometric sine or cosine function.
Simple Harmonic Motion and 17 the Reference Circle Simple harmonic motion, like any motion, can be described in terms of displacement, velocity, and acceleration.
DISPLACEMENT 18
For any object in simple harmonic motion, the time required to complete one cycle is the period, T (MKS) 19 Instead of the period, it is more convenient to speak of the angular frequency w of the motion, the frequency being just the number of cycles of the motion per second.
One cycle per second is referred to as one hertz (Hz). One thousand cycles per second is called one kilohertz (khz). 20 w is often called the angular frequency.
FREQUENCY OF VIBRATION 21 F = ma k Acoswt m Aw 2 cos wt w k m w in rad / s
Body Mass Measurement Device Astronauts who spend long periods of time in orbit periodically measure their body masses as part of their health-maintenance programs. On earth, it is simple to measure body weight W with a scale and convert it to mass m using the acceleration due to gravity, since W = mg. However, this procedure does not work in orbit, because both the scale and the astronaut are in free-fall and cannot press against each other. Instead, astronauts use a body mass measurement device. This device consists of a spring-mounted chair in which the astronaut sits. The chair is then started oscillating in simple harmonic motion. The period of the motion is measured electronically and is automatically converted into a value of the astronaut s mass, after the mass of the chair is taken into account. The spring used in one such device has a spring constant of 606 N/m, and the mass of the chair is 12.0 kg. The measured oscillation period is 2.41 s. Find the mass of the astronaut.
w 2 T k m 23
Velocity in SHM v (-) m v (+) x = -A x = 0 x = +A Velocity is positive when moving to the right and negative when moving to the left. It is zero at the end points and a maximum at the midpoint in either direction (+ or -).
Velocity as Function of Position. x a m v x = -A x = 0 x = +A 1 mv 2 1 kx 2 1 ka 2 2 2 2 k v A x m 2 2 v max when x = 0: v k m A
Check Your Understanding 2 26 The drawing shows plots of the displacement x versus the time t for three objects undergoing simple harmonic motion. Which object, I, II, or III, has the greatest maximum velocity? II
Example 5: A 2-kg mass hangs at the end of a spring whose constant is k = 800 N/m. The mass is displaced a distance of 10 cm and released. What is the velocity at the instant the ½mv displacement 2 + ½kx 2 = is ½kA x = 2 +6 cm? v k v A x m 2 2 800 N/m (0.1 m) (0.06 m) 2 kg v = ±1.60 m/s 2 2 m +x
Example 5 (Cont.): What is the maximum velocity for the previous problem? (A = 10 cm, k = 800 N/m, m = 2 kg.) The velocity is maximum when x = 0: 0 ½mv 2 + ½kx 2 = ½kA 2 v k m A 800 N/m (0.1 m) 2 kg m +x v = ± 2.00 m/s
Example 3. The Maximum 29 Speed of a Loudspeaker Diaphragm The diaphragm of a loudspeaker moves back and forth in simple harmonic motion to create sound. The frequency of the motion is f = 1.0 khz and the amplitude is A = 0.20 mm. (a)what is the maximum speed of the diaphragm? (b)where in the motion does this maximum speed occur?
(a) 30 (b) The speed of the diaphragm is zero when the diaphragm momentarily comes to rest at either end of its motion: x = +A and x = A. Its maximum speed occurs midway between these two positions, or at x = 0 m.
Acceleration in SHM +a -x +x -a m x = -A x = 0 x = +A Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.) F ma kx Acceleration is a maximum at the end points and it is zero at the center of oscillation.
Acceleration vs. Displacement x a m v x = -A x = 0 x = +A Given the spring constant, the displacement, and the mass, the acceleration can be found from: F ma kx Note: Acceleration is always opposite to displacement. or a kx m
Example 3: A 2-kg mass hangs at the end of a spring whose constant is k = 400 N/m. The mass is displaced a distance of 12 cm and released. What is the acceleration at the instant the displacement is x = +7 cm? (400 N/m)(+0.07 m) kx a 2 kg a kx m a = -14.0 m/s 2 a m +x Note: When the displacement is +7 cm (downward), the acceleration is -14.0 m/s 2 (upward) independent of motion direction.
Example 4: What is the maximum acceleration for the 2-kg mass in the previous problem? (A = 12 cm, k = 400 N/m) The maximum acceleration occurs when the restoring force is a maximum; i.e., when the stretch or compression of the spring is largest. F = ma = -kx x max = A a ka 400 N( 0.12 m) m 2 kg Maximum Acceleration: m a max = ± 24.0 m/s 2 +x
Example 5. 35 The Loudspeaker Revisited The Maximum Acceleration A loudspeaker diaphragm is vibrating at a frequency of f = 1.0 khz, and the amplitude of the motion is A = 0.20 mm. (a)what is the maximum acceleration of the diaphragm, and (b)where does this maximum acceleration occur?
36 (a) (b) the maximum acceleration occurs at x = +A and x = A
Hooke s Law When a spring is stretched, there is a restoring force that is proportional to the displacement. F = -kx x m F The spring constant k is a property of the spring given by: k = DF Dx
Ch10. Simple Harmonic Motion and Elasticity The Ideal Spring and Simple Harmonic Motion F Applied kx The constant k is called the spring constant A spring that F Applied behaves kx according to said to be an ideal spring. is 38
mg = kd 0, which gives d 0 = mg/k. The restoring force 39 also leads to simple harmonic motion when the object is attached to a vertical spring, just as it does when the spring is horizontal. When the spring is vertical, however, the weight of the object causes the spring to stretch, and the motion occurs with respect to the equilibrium position of the object on the stretched spring.
Example 1. A Tire Pressure Gauge In a tire pressure gauge, the air in the tire pushes against a plunger attached to a spring when the gauge is pressed against the tire valve. Suppose the spring constant of the spring is k = 320 N/m and the bar indicator of the gauge extends 2.0 cm when the gauge is pressed against the tire valve. What force does the air in the tire apply to the spring? 40
Conceptual Example 2. Are Shorter Springs Stiffer Springs? 41 A 10-coil spring that has a spring constant k. If this spring is cut in half, so there are two 5-coil springs, what is the spring constant of each of the smaller springs? Shorter springs are stiffer springs. Sometimes the spring constant k is referred to as the stiffness of the spring, because a large value for k means the spring is stiff, in the sense that a large force is required to stretch or compress it.
Example 2: A 4-kg mass suspended from a spring produces a displacement of 20 cm. What is the spring constant? The stretching force is the weight (W = mg) of the 4-kg mass: 20 cm F = (4 kg)(9.8 m/s 2 ) = 39.2 N Now, from Hooke s law, the force constant k of the spring is: m F DF k = = Dx 39.2 N 0.2 m k = 196 N/m
Check Your Understanding 1 43 A 0.42-kg block is attached to the end of a horizontal ideal spring and rests on a frictionless surface. The block is pulled so that the spring stretches by 2.1 cm relative to its unstrained length. When the block is released, it moves with an acceleration of 9.0 m/s 2. What is the spring constant of the spring? 180 N/m
44 kx = ma 2.1c m k 2.1 100 0.42 9.0m / s 2 k 0.42 9.0 2.1 100 180N / m