Presentation on theme: "Kinematics of simple harmonic motion (SHM)"— Presentation transcript:
1 Kinematics of simple harmonic motion (SHM) Physics 12Resource: Giancoli Chapter 11Kinematics of simple harmonic motion (SHM)
2 Objectives Describe examples of oscillations Define the terms displacement, amplitude, frequency, period and phase difference.Define simple harmonic motion (SHM).Solve problems using the defining equation for SHM.
3 Objectives Apply equations for the kinematics of SHM. Solve problems both graphically and by calculation, for acceleration, velocity and displacement during SHM.
4 Waves and oscillations To oscillate means to move back and forth.Can you give examples of oscillation?
5 Waves and oscillations All things that oscillate / vibrate are ultimately linked.Their motion can be explained using the concept of waves.For simplicity, let us take the example of a simple pendulum.
6 Kinematics of SHM Consider the following pendulum: A block of mass m attached to a spring that oscillates horizontally on a frictionless surface.The equilibrium position is the point at which the mass rests, without external stretching or compression.
7 Kinematics of SHMThe force exerted by the spring is represented by the following expression:F = -kxwhere F is the force exerted by the springk is the spring constant (dependent on the material of the spring)and x is the displacement of the mass mWhat does the negative sign signify?
8 Kinematics of SHMThe negative sign connotes that the restorative force of a spring is always in the opposite direction of the displacement.When the spring is stretched and displacement x is to the right, the spring exerts a force that restores it to the left (back to equilibrium position x = 0)When the spring is compressed and displacement is to the left, the spring exert a force to the right.
9 Kinematics of SHM Consider the following simple pendulum: A mass m hanging vertically from a spring with spring constant k.Would the equilibrium position x0 be the same as the pendulum which oscillates horizontally?
10 Kinematics of SHMThe spring would be stretched an extra amount related to the weight of the mass:F = mgThe equilibrium point may be defined as the point whereΣF = 0ΣF = mg – kx00 = mg – kx0
11 Kinematics of SHMA family of 200 kg steps into a 1200-kg car and the car lowers 3.0 cm.What is the spring constant k of the car’s springs?How much further would the car lower if the family was 300 kg?
13 Definition of SHMWhen the family’s mass is 200 kg, the springs compress 3.0 cmWhen half of the mass is added, i.e. the family’s mass is 300 kg, the springs compress 4.5 cm.What do you notice?
14 Definition of SHMAn oscillator for which the force exerted is proportional to its displacement is called a simple harmonic oscillator.In other words, simple harmonic motion (SHM) is a type of motion for whichF = -kx holds true.
15 Definition of SHMAre the following oscillators simple harmonic in nature?F = 0.5x2F = -2.3yF = 8.6xF = -40t
22 Sinusoidal nature of SHM Definition of termsTermDefinitiondisplacement, xthe movement of particles above and below the mean positionamplitude, Athe difference between the maximum displacement and the mean positionperiod, Tthe amount of time required to complete one cycle (between two identical positions)frequency, fthe number of complete cycles passing a given point in one second (Hz)wavelength, λthe distance covered in a complete wave cycle.
24 Sinusoidal nature of SHM Construct the following graphs: displacement-time velocity – time acceleration – time for a pendulum starting at maximum displacement and one starting at equilibrium position.
25 Sinusoidal nature of SHM SHM is said to be sinusoidal in nature.Depending on the starting point, the relationship between certain variables (displacement, velocity, acceleration) and time can either be a sine or cosine function.
26 Sinusoidal nature of SHM Relationship between period T and frequency f f (in s-1 or Hz) = T (in s) = What is the relationship between period and frequency?
27 Sinusoidal nature of SHM Period T of SHM T = 2π
28 Sinusoidal nature of SHM A spider of mass 0.30 g waits in its web of negligible mass. A slight movement causes the web to vibrate with a frequency of about 15 Hz.Estimate the value of the spring stiffness constant k for the webAt what frequency would you expect the web to vibrate if an insect of mass 0.10 g were trapped with the spider.
30 Energy of SHM Is work done when a spring is stretched or compressed? How is the energy stored?
31 Energy of SHMWhen a spring is stretched or compressed, work is done and converted into the potential energy of the spring.Elastic potential energy is given by the expression:PE = ½ kx2Describe what happens to a spring in terms of energy as it completes one cycle.
32 Energy of SHMTotal mechanical E = KE +PE Total mechanical E = ½ mv2 + ½ kx2 Derive expressions for total mechanical energy, E, at maximum displacement and equilibrium position.
33 Energy of SHMAt x = A and x = -A, v = 0, therefore: E = ½ m(02) + ½ kA2 E = ½ kA2 At equilibrium point, x = 0 and v = vmax, therefore: E = ½ mvmax2 + ½ k(0)2 E = ½ mvmax2
34 Energy of SHMUse the conservation of mechanical energy to deduce an expression for the instantaneous velocity of an SHO (velocity v at any time) in terms of vmax, x, and A: v = ± vmax
35 Energy of SHMSuppose a spring oscillator is stretched to twice the amplitude (x = 2A). What happens to the:energy of the systemmaximum velocity of the oscillatormaximum acceleration of the mass
36 Energy of SHM energy is quadrupled maximum velocity is doubled acceleration is doubled
37 Energy of SHMA spring stretches m when a kg mass is gently lowered on it. The spring is set up on a frictionless table. The mass is pulled so that the spring is stretched m from the equilibrium point then released from rest. Determine the:spring stiffness constant kamplitude of horizontal oscillation Amagnitude of maximum velocity vmaxmagnitude of velocity v when the mass is m from equilibrium andmagnitude of the maximum acceleration amax of the mass
38 Energy of SHM k = 19.6 N/m A = 0.100 m vmax = 0.808 m/s v = 0.70 m/s amax = 6.53 m/s2
39 Simple pendulumConsider this simple pendulum. Does it oscillate? Does F = - kx still apply?
40 Simple pendulum Derive expressions for: displacement along the arc the restoring force F tangent to the arc