Simple Harmonic Motion Wenny Maulina

Simple harmonic motion Simple harmonic motion (SHM) f=w/(2p) Solution: Acosf What is SHM? t=-f/w t=0 A simple harmonic motion is the motion of an oscillating system which satisfies the following condition: Motion is about an equilibrium position at which point no net force acts on the system. The restoring force is proportional to and oppositely directed to the displacement. 3. Motion is periodic. w=w0; w=2w0 ; w=3w0 By Dr. Dan Russell, Kettering University

Simple Harmonic Motion, SHM Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. x F A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx

Oscillations of a Spring Hooke’s Law states Fs = -kx Fs is the restoring force. It is always directed toward the equilibrium position. Therefore, it is always opposite the displacement from equilibrium. k is the force (spring) constant. x is the displacement.

Oscillations of a Spring In a, the block is displaced to the right of x = 0. The position is positive. The restoring force is directed to the left (negative). In b, the block is at the equilibrium position. x = 0 The spring is neither stretched nor compressed. The force is 0. In c, the block is displaced to the left of x = 0. The position is negative. the right (positive).

Example 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 length. When the block is released, it moves with an acceleration of 9.0 m/s2. What is the spring constant of the spring?

2.1cm kx = ma

Displacement in SHM m x = 0 x = +A 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.

Velocity in SHM m x = 0 x = +A 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.

Acceleration in SHM m +a -a x = 0 x = +A x = -A Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.) Acceleration is a maximum at the end points and it is zero at the center of oscillation.

Acceleration vs. Displacement x v a m x = 0 x = +A x = -A Given the spring constant, the displacement, and the mass, the acceleration can be found from: or Note: Acceleration is always opposite to displacement.

Simple harmonic motion Displacement, velocity and acceleration in SHM Displacement Velocity Acceleration

Example 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. What is the maximum speed of the diaphragm? Where in the motion does this maximum speed occur?

(a) (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.

Example A loudspeaker diaphragm is vibrating at a frequency of f = 1.0 kHz, and the amplitude of the motion is A = 0.20 mm. What is the maximum acceleration of the diaphragm, and where does this maximum acceleration occur?

(a) (b) the maximum acceleration occurs at x = +A and x = –A

Example 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?

Example The cone of a loudspeaker oscillates in SHM at a frequency of 262 Hz. The amplitude at the center of the cone is A = 1.5 x 10-4 m, and at t = 0, x = A. (a) What equation describes the motion of the center of the cone? (b) What are the velocity and acceleration as a function of time? (c) What is the position of the cone at t = 1.00 ms (= 1.00 x 10-3 s)? Figure 14-9. Caption: Example 14-4. A loudspeaker cone. Solution: a. The angular frequency is 1650 rad/s, so x = (1.5 x 10-4m)cos(1650t). b. The maximum velocity is 0.25 m/s, so v = -(0.25 m/s)sin(1650t). The maximum acceleration is 410 m/s2, so a = -(410 m/s2)cos(1650t). c. At t = 1 ms, x = -1.2 x 10-5 m.

Solution:

Solution:

Example A spring stretches 0.150 m when a 0.300-kg mass is gently attached to it. The spring is then set up horizontally with the 0.300-kg mass resting on a frictionless table. The mass is pushed so that the spring is compressed 0.100 m from the equilibrium point, and released from rest. Determine: (a) the spring stiffness constant k and angular frequency ω; (b) the amplitude of the horizontal oscillation A; (c) the magnitude of the maximum velocity vmax; (d) the magnitude of the maximum acceleration amax of the mass; (e) the period T and frequency f; (f) the displacement x as a function of time; and (g) the velocity at t = 0.150 s. Solution. A. The spring constant is 19.6 N/m, so ω = 8.08/s−1. b. Since the spring is released from rest, A = 0.100 m. c. The maximum velocity is 0.808 m/s d. The maximum acceleration occurs when the displacement is maximum, and equals 6.53 m/s2. e. The period is 0.777 s, and the frequency is 1.29 Hz. f. At t = 0, x = -A; this is a negative cosine. X = (0.100 m)cos (8.08t – π). g. V = dx/dt; at 0.150 s it is 0.756 m/s.

Solution:

Oscillations of a Spring k m F = - k x x x F = - k x

Simple Pendulum Displacement : x = L θ Returning force : F = - mg sin θ Newton Law : F = m d2θ/dt2 Taylor series : sin 𝜃 = 𝜃 − 𝜃 3 3! + 𝜃 5 5! −⋯ Small oscillation approximation : sin θ ≈ θ Acceleration : d2θ/dt2 = - (g/L) x Solutions : Y = A sin wt , where w2 = g/L Y = A cos wt Y = A eiwt

Energy in SHM The fact that the velocity is zero at maximum displacement in simple harmonic motion and is a maximum at zero displacement illustrates the important concept of an exchange between kinetic and potential energy. If no energy is dissipated then all the potential energy becomes kinetic energy and vice versa, so that the values of (a) the total energy at any time, (b) the maximum potential energy and (c) the maximum kinetic energy will all be equal; that is

Energy in SHM No friction BTW: w2 Energy conservation Energy conservation in a SHM No friction BTW: w2

Energy in SHM kinetic energy E energy energy Energy conservation in a SHM kinetic energy E energy energy distance from equilibrium point Time potential energy

Energy in SHM This graph shows the potential energy function of a spring. The total energy is constant. Figure 14-11. Caption: Graph of potential energy, U = ½ kx2. K + U = E = constant for any point x where –A ≤ x ≤ A. Values of K and U are indicated for an arbitrary position x.

Example An object of mass m = 0.200 kg that is vibrating on a horizontal frictionless table. The spring has a spring constant k = 545 N/m. It is stretched initially to x0 = 4.50 cm and then released from rest (see part A of the drawing). Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf = 2.25 cm and (b) xf = 0 cm.

(a) Since x0 = 0.0450 m and xf = 0.0225 m, (b) When x0 = 0.0450 m and xf = 0 m,

Example What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)? L

Example 𝑇 2 =4 𝜋 2 𝐿 𝑔 L= 𝑇 2 𝑔 4 𝜋 2 L= 2 2 9,8 4 𝜋 2 What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)? 𝑇 2 =4 𝜋 2 𝐿 𝑔 L= 𝑇 2 𝑔 4 𝜋 2 L= 2 2 9,8 4 𝜋 2 L L = 0.993 m

Example Many tall building have mass dampers, which are anti-sway devices to prevent them from oscillating in a wind. The device might be a block oscillating at the end of a spring and on a lubricated track. If the building sways, say eastward, the block also moves eastward but delayed enough so that when it finally moves, the building is then moving back westward. Thus, the motion of the oscillator is out of step with the motion of the building. Suppose that the block has mass m = 2.72 x 105 kg and is designed to oscillate at frequency f = 10.0 Hz and with amplitude xm = 20.0 cm. (a) What is the total mechanical energy E of the spring-block system?

Damped Oscillation The friction reduces the mechanical energy of the system as time passes, and the motion is said to be damped. Damped harmonic motion is harmonic motion with a frictional or drag force. There are systems in which damping is unwanted, such as clocks and watches. Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.

Damped Oscillation 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.

Damped Oscillation

Forced Oscillation When a periodically varying driving force with angular frequency is applied to a damped harmonic oscillator, the resulting motion is called a forced oscillation. There are two frequencies involved in a forced oscillator: ω0, the natural angular frequency of the oscillator, without the presence of any external force, and ω, the angular frequency of the applied external force. If the frequency is the same as the natural frequency, the amplitude can become quite large. This is called resonance.

Forced Oscillation The sharpness of the resonant peak depends on the damping. If the damping is small (A) it can be quite sharp; if the damping is larger (B) it is less sharp. Figure 14-23. Caption: Resonance for lightly damped (A) and heavily damped (B) systems. Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.

ASSIGNMENT 1 Should Be Submitted Next Week Please make a paper about Standing waves and Travelling waves