Phys 1022: Introduction, Pg 1 Physics 1022: Chapter 14 Waves Anatomy of a wave Simple harmonic motion Energy and simple harmonic motion Phys 1022: Introduction, Pg 2 Page 1 1 Waves New Topic Phys 1022: Introduction, Pg 3 Simple Harmonic Motion: The restoring force is proportional to the negative of the displacement (like F=-kx) ma = F m d 2x = −kx dt 2 d 2x k + x=0 dt 2 m d 2x k + ω 2 x = 0, with ω = 2 dt m general solution: x = A cos(ωt + φ ) A is amplitude, φ is phase angle. They are determined by initial conditions (the value of x and v at t=0.) Phys 1022: Introduction, Pg 4 Page 2 2 Phase angle x = A cos(ωt + φ ) v= dx = − Aω sin(ωt + φ ) dt A measure of different starting positions (and velocities) Both A and φ can be determined by initial conditions: At t = 0 : x0 = A cosφ , v0 = −ωA sin φ A= 2 x02 + (v0 / ω ) , tanφ = − v0 x0ω Phys 1022: Introduction, Pg 5 The graph shows a particle in SHM. (a) What is the phase constant φ0? (b) What is the phase of the particle at the numbered points on the graph? (c) Place dots (with labels) on the circle to indicate the particle s position corresponding to the numbered points. Phys 1022: Introduction, Pg 6 Page 3 3 What does the v vs. t graph look like? A particle moves CCW around a circle at constant speed. From the phase constants, show the particle s initial position and sketch two cycles of the x vs. t graph. Phys 1022: Introduction, Pg 7 Simple Harmonic ↔ Circular Motion If we look at uniform circular motion from the side, the object appears to move in simple harmonic motion y Top View: v = v0 vx = v0 Consider the x-position of the object: x = Acos θ A now θ = ω t θ x v = v0 vx = 0 Side View: x x = Acos (ωt) Click here for demo y x = +A Phys 1022: Introduction, Pg 8 Page 4 4 ConcepTest 4(Post)Energy in SHM This is the potential energy diagram of a particle oscillating on a spring. What is the equilibrium length of the spring? 1. 12 cm 2. 16 cm 3. 20 cm 4. 24 cm 5. 28 cm 6. cannot be determined from the graph Phys 1022: Introduction, Pg 9 ConcepTest 5(Post)Energy in SHM If the particle s turning points are 14 cm and 26 cm, draw a line that indicates the total energy and then determine the particle s maximum kinetic energy. 1. 2.5 J 2. 5 J 3. 7.5 J 4. 10 J 5. 12.5 J 6. 15 J 7. other Etot ~ 6.5 J Phys 1022: Introduction, Pg 10 Page 5 5 Energy in Simple Harmonic Motion Energy of the oscillating system at any time is constant: Etotal = PEspring + KEmass k ENERGY IS CONSERVED xmax = A m Etotal = PEspring = 1/2kA2 x Frestore k m at any point in between … Etotal = 1/2kx2 + 1/2mv2 Frestore m at end, x = A and v = 0 (KE = 0) at x = 0 , PE = 0 and v = v0 Etotal = KEmass = 1/2mv02 equilibrium position Phys 1022: Introduction, Pg 11 1 2 2 kA 2 = 12 mvmax What will be the turning points if the particle s total energy is doubled? Draw a graph of the particle s kinetic energy as a function of position. Phys 1022: Introduction, Pg 12 Page 6 6 ConcepTest 6(Post)Simple Harmonic Motion A block oscillating on a spring has a period of T = 4 s. If the mass of the block is halved, what is the new period? 1. 1 s 2. 2 s 3. 2.8 s 4. 4 s 5. 5.6 s 6. 8 s 7. 16 s Phys 1022: Introduction, Pg 13 ConcepTest 7(Post)Simple Harmonic Motion A block oscillating on a spring has a period of T = 4 s. If the spring constant is quadrupled, what is the new period? 1. 1 s 2. 2 s 3. 2.8 s 4. 4 s 5. 5.6 s 6. 8 s 7. 16 s Phys 1022: Introduction, Pg 14 Page 7 7 ConcepTest 8(Post)Simple Harmonic Motion A block oscillating on a spring has a period of T = 4 s. If the oscillation amplitude is doubled, what is the new period? 1. 1 s 2. 2 s 3. 2.8 s 4. 4 s 5. 5.6 s 6. 8 s 7. 16 s Phys 1022: Introduction, Pg 15 The graph shows x vs. t for a particle in SHM. (a) Draw the v vs. t and the a vs. t graphs. (b) When x is greater than zero, is a ever greater than zero? When? (c) When x is greater than zero, is v ever greater than zero? When? Can we describe the entire motion of this oscillating system? Phys 1022: Introduction, Pg 16 Page 8 8 Based on this reference graph of x vs. t, draw the new graphs that represent the following conditions: (a) amplitude and frequency are doubled (b) amplitude is halved and mass is quadrupled (c) phase constant is increased by π/2 rad (d) max. speed is doubled while amplitude stays constant Phys 1022: Introduction, Pg 17 Simple Pendulum New Topic Phys 1022: Introduction, Pg 18 Page 9 9 ConcepTest 9(Post)Simple Pendulum A pendulum on Planet X, where the value of g is unknown, oscillates with a period of T = 4 s. If the mass is quadrupled, what is the new period? 1. 1 s 2. 2 s 3. 2.8 s 4. 4 s 5. 5.6 s 6. 8 s 7. 16 s Phys 1022: Introduction, Pg 19 The Simple Pendulum Consider only small oscillations ⇒ sinθ ≈ θ (Try this on your calculator, but θ must be in radians!) Restoring force: Spring Oscillator F = –kx T = 2π √ m / k F = –mg sin θ = - mg θ = –mg x/L = –(mg/L) x Pendulum F = –mg/L x T = 2π √ m / (mg/L) T = 2π √ L / g Period of a pendulum does NOT depend on ý mass ý amplitude Phys 1022: Introduction, Pg 20 Page 10 10 ConcepTest 12(Post)Simple Pendulum If a pendulum with period T on Earth is taken to the Moon, how will the period change? 1. increases 2. decreases 3. stays the same 4. no way to tell Phys 1022: Introduction, Pg 21 ConcepTest 13(Post)Simple Pendulum If a mass-spring system with period T on Earth is taken to the Moon, how will the period change? 1. increases 2. decreases 3. stays the same 4. no way to tell Phys 1022: Introduction, Pg 22 Page 11 11 ConcepTest 14(Post)Simple Pendulum A pendulum in an elevator has period T when the elevator is at rest. If the elevator is accelerating upward, how will the period change? 1. increases 2. decreases 3. stays the same 4. no way to tell What happens to the period of the pendulum if the elevator is in free fall? Phys 1022: Introduction, Pg 23 ConcepTest 15(Post)To the Center of the Earth A hole is drilled through the center of the Earth and emerges on the other side. You jump into the hole. What happens to you? 1. 2. 3. 4. 5. you fall to the center and stop you go all the way through and continue off into space you fall to the other side of the Earth and stay there you fall to the other side of the Earth and then return you won t fall at all Phys 1022: Introduction, Pg 24 Page 12 12

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