2. Topic 4.1 Extended B – Pendulum system SHM  In its simplest form, a pendulum is a mass hanging from a string.  The mass is called the pendulum bob.  We initially displace the bob from the vertical by an angle θ m. θmθm  We then release it, and watch it oscillate in harmonic motion.  We call a mass on a string a simple pendulum. The name is BOB. Pendulum Bob.

3. Topic 4.1 Extended B – Pendulum system SHM T HE S IMPLE P ENDULUM  Suppose during its oscillation the string makes an arbitrary angle θ with the vertical. θmθm θ  At this instant two forces act on the bob: gravity and tension. mg T  Furthermore, the bob moves along the arc of a circle of radius ℓ, the length of the string. ℓ

4. Topic 4.1 Extended B – Pendulum system SHM T HE S IMPLE P ENDULUM mg T θ  Since the mass is moving in a circle, we will use torque. Σ τ = Iα  If we let clockwise be positive we have ℓT sin 0°+ ℓmg sin θ = Iα 0 Iα = ℓmg sin θ FYI: The torque is a restoring torque, since it opposes the angular displacement θ. Thus we place the minus sign in the equation. (Recall F = -kx for the spring.) mℓ 2 α = -ℓmg θ FYI: Recall that sin θ = θ (for small enough θ), because sin θ = θ - θ 3 /3! + θ 5 /5! - … α = - θ gℓgℓ θ” = - θ gℓgℓ θ” = -ω 2 θ Where ω = gℓgℓ Why?  Clearly, the previous equation will have solutions θ(t) = θ m sin(ωt + φ) θ(t) = θ m cos(ωt + φ) Where ω = gℓgℓ The simple pendulum Question: What happened to the I?

5. Topic 4.1 Extended B – Pendulum system SHM T HE S IMPLE P ENDULUM  What length of string would you use so the simple pendulum had a period of 2 seconds?  Since ω = gℓgℓ and T = 2πω2πω T = 2π ℓgℓg then Simple pendulum  Thus ℓ = g T2πT2π 2 = 10 22π22π 2 = 1.0132 m

6. Topic 4.1 Extended B – Pendulum system SHM T HE P HYSICAL P ENDULUM  If, instead of a mass on a string we have an extended object which hangs from one end, we have a physical pendulum.  For example, a meter stick hanging from a hole located 20 cm from its end is a physical pendulum.  After analyzing the motion of a general physical pendulum we’ll come back to the meter stick example.

7. Topic 4.1 Extended B – Pendulum system SHM T HE P HYSICAL P ENDULUM  Consider the generalized physical pendulum.  Point O is the axis, and Point C is the cm.  The distance between O and C is h. c o h  The tension along the line h will not contribute to the torque, but the weight will. mg θ hT sin 0°+ hmg sin θ = Iα 0 Iα = hmg sin θ Σ τ = Iα FYI: The weight acts from the cm. α = - θ mgh I θ” = - θ mgh I θ” = -ω 2 θ Where ω = mgh I Why? FYI: Don’t forget: I = I cm + mh 2 Why?

8. Topic 4.1 Extended B – Pendulum system SHM T HE P HYSICAL P ENDULUM  To summarize the physical pendulum: Where ω = mgh I and I = I cm + mh 2 θ(t) = θ m sin(ωt + φ) θ(t) = θ m cos(ωt + φ) Physical pendulum  Since ω = mgh I and T = 2πω2πω then Physical pendulum T = 2π I mgh

9. Topic 4.1 Extended B – Pendulum system SHM T HE P HYSICAL P ENDULUM  Returning to the ruler problem, what is the predicted period? T = 2π I mgh T = 2π I cm + mh 2 mgh T = 2π 1 2 + 0.30 2 9.8(.30) 1 12 T = 2π mL 2 + mh 2 mgh 1 12 T = 1.526 s