Topic 4.1 Extended B – Pendulum system SHM

1. θm Topic 4.1 ExtendedB – 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 θm. We then release it, and watch it oscillate in harmonic motion. The name is BOB. Pendulum Bob. We call a mass on a string a simple pendulum.

2. θ T θm mg Topic 4.1 ExtendedB – Pendulum system SHM THE SIMPLE PENDULUM Suppose during its oscillation the string makes an arbitrary angle θwith the vertical. At this instant two forces act on the bob: gravity and tension. Furthermore, the bob moves along the arc of a circle of radius ℓ, the length of the string. ℓ

3. FYI: Recall that sin θ=θ(for small enough θ), because sin θ=θ-θ3/3! + θ5/5! - … 0 T g ℓ g ℓ α = - θ θ” = - θ mg g ℓ g ℓ The simple pendulum Where ω = Where ω = 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.) Topic 4.1 ExtendedB – Pendulum system SHM THE SIMPLE PENDULUM Since the mass is moving in a circle, we will use torque. If we let clockwise be positive we have Στ = Iα ℓTsin0°+ ℓmgsinθ = Iα θ Iα = ℓmgsinθ Question: What happened to the I? mℓ2α = -ℓmgθ Why? θ” = -ω2θ Clearly, the previous equation will have solutions θ(t) = θmsin(ωt + φ) θ(t) = θmcos(ωt + φ)

4. Simple pendulum ℓ g T = 2π 2π ω 2 T 2π T = ℓ = g 2 2 2π = 10 g ℓ ω = Topic 4.1 ExtendedB – Pendulum system SHM THE SIMPLE PENDULUM What length of string would you use so the simple pendulum had a period of 2 seconds? Since then and Thus = 1.0132 m

5. Topic 4.1 ExtendedB – Pendulum system SHM THE PHYSICAL PENDULUM 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.

6. h o 0 c mgh I mgh I θ” = - θ α = - θ mg mgh I Where ω = Topic 4.1 ExtendedB – Pendulum system SHM THE PHYSICAL PENDULUM FYI: The weight acts from the cm. Consider the generalized physical pendulum. Point O is the axis, and Point C is the cm. The distance between O and C is h. The tension along the line h will not contribute to the torque, but the weight will. θ Στ = Iα hTsin0°+ hmgsinθ = Iα Iα = hmgsinθ Why? Why? FYI: Don’t forget: I = Icm+ mh2 θ” = -ω2θ

7. Physical pendulum 2π ω T = ω = Physical pendulum mgh I mgh I Where ω = I mgh T = 2π Topic 4.1 ExtendedB – Pendulum system SHM THE PHYSICAL PENDULUM To summarize the physical pendulum: θ(t) = θmsin(ωt + φ) θ(t) = θmcos(ωt + φ) and I = Icm+ mh2 Since then and

8. 1 12 mL2 + mh2 mgh Icm+ mh2 mgh T = 2π T = 2π 1 12 12 + 0.302 9.8(.30) T = 2π I mgh T = 2π Topic 4.1 ExtendedB – Pendulum system SHM THE PHYSICAL PENDULUM Returning to the ruler problem, what is the predicted period? T = 1.526 s