Download Simple Harmonic Motion - General Physics I - Lecture Slides and more Slides Physics in PDF only on Docsity! Simple Harmonic Motion and Uniform Circular Motion • A ball is attached to the rim of a turntable of radius A • The focus is on the shadow that the ball casts on the screen • When the turntable rotates with a constant angular speed, the shadow moves in simple harmonic motion Docsity.com Period and Frequency from Circular Motion • Period • This gives the time required for an object of mass m attached to a spring of constant k to complete one cycle of its motion • Frequency • Units are cycles/second or Hertz, Hz • The angular frequency is related to the frequency k m2T π= m k 2 1 T 1ƒ π == m kƒ2 =π=ω Docsity.com 8 x t( ) = Acosωt v t( ) = Δx Δt = −Aω sinωt a t( ) = Δv Δt = −Aω 2 cosωt Docsity.com 9 Example: The period of oscillation of an object in an ideal mass-spring system is 0.50 sec and the amplitude is 5.0 cm. What is the speed at the equilibrium point? Idea: let’s use energy conservation: at equilibrium x = 0: 222 2 1 2 1 2 1 mvkxmvUKE =+=+= Since E=constant, at equilibrium (x = 0) the KE must be a maximum. Thus, v = vmax = Aω. ( )( ) cm/sec 8.62rads/sec 6.12cm 5.0 and rads/sec 6.12 s 50.0 22 === === Aωv T ππ ω Docsity.com Verification of Sinusoidal Nature • This experiment shows the sinusoidal nature of simple harmonic motion • The spring mass system oscillates in simple harmonic motion • The attached pen traces out the sinusoidal motion Docsity.com Simple Pendulum • In general, the motion of a pendulum is not simple harmonic • However, for small angles, it becomes simple harmonic • In general, angles < 15° are small enough so sin θ = θ F = - m g θ • This force law looks like Hooke’s Law! Docsity.com Simple Pendulum Compared to a Spring-
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15 The Simple Pendulum (1) sin xx maL TxTF =−=−=∑ θ Fy∑ = T cosθ −mg =may (2) When θ is small, mgT ay =⇒ ≈= (2)Equation 0 , 1cosθ x L gama L mgx xx −=⇒=⇒ - )1(Equation Motion is Simple Harmonic: For small oscillations xax 2ω−= L g =ω g LT π2= Docsity.com 18 Example: The gravitational potential energy of a pendulum is U = mgy. Taking y = 0 at the lowest point of the swing, show that y = L(1-cosθ). θ L y=0 L Lcosθ )cos1( θ−= Ly Docsity.com Example: pendulum clock A pendulum clock that works perfectly on Earth is taken to the Moon. (a) Does it run fast or slow there? (b) If the clock is started at 12:00 midnight, what will it read after one Earth-day (24.0 h)? Assume that the free-fall acceleration on the Moon is 1.63 m/s2. Docsity.com 20 A physical pendulum is any rigid object that is free to oscillate about some fixed axis. The period of oscillation of a physical pendulum is not necessarily the same as that of a simple pendulum. Physical Pendulum Docsity.com 23 Forced Oscillations and Resonance A force can be applied periodically to a damped oscillator (a forced oscillation). When the force is applied at the natural frequency of the system, the amplitude of the oscillations will be a maximum. This condition is called resonance. Docsity.com Let’s watch a movie! Docsity.com
