- #1
Wavefunction
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- 4
Homework Statement
The pendulum of a grandfather clock activates an escapement mechanism every time it passes
through the vertical. The escapement is under tension (provided by a hanging weight) and gives the
pendulum a small impulse a distance [itex] l [/itex] from the pivot. The energy transferred by this impulse
compensates for the energy dissipated by friction, so that the pendulum swings with a constant
amplitude.
a) What is the impulse needed to sustain the motion of a pendulum of length [itex] L [/itex] and mass [itex] m [/itex], with
an amplitude of swing [itex] θ_0 [/itex] and quality factor [itex] Q [/itex]? You can assume [itex] Q [/itex] is large and [itex] θ_0 [/itex] is small
Homework Equations
[itex] \ddot{θ}+2β\dot{θ}+(ω_0)^2θ=0 [/itex] where [itex] θ(t)=exp[-βt][θ_0cos(ω_1t)+\frac{βθ_0}{ω_1}sin(ω_1t)] [/itex] from initial conditions [itex] θ(0)=θ_0 [/itex] and [itex] \dot{θ(0)}=0 [/itex]
[itex] (ω_0)^2 \equiv \frac{mgL}{I}, I =mL^2, 2β \equiv \frac{b}{I} [/itex]
The Attempt at a Solution
Okay so first I calculated the initial energy: [itex] E_i = \frac{1}{2}m(ω_0)^2L^2[θ_0]^2 [/itex]
Next I calculated the energy at a time [itex] t_0 [/itex] later: [itex] E_f = \frac{1}{2}mL^2[\dot{θ(t_0)}]^2 [/itex]
Then I took the change in energy: [itex] ΔE_- = E_f-E_i = \frac{1}{2}mL^2[[\dot{θ(t_0)}]^2-(ω_0θ_0)^2] [/itex]
In order to offset this change in energy I need to add an energy [itex] ΔE_+ [/itex] such that [itex] ΔE_-=ΔE_+ [/itex].
So I'll let [itex] ΔE_+ = \frac{1}{2}ml^2[Δ\dot{θ}]^2 [/itex]
Since [itex] ΔE_-=ΔE_+ → \frac{1}{2}mL^2[[\dot{θ(t_0)}]^2-(ω_0θ_0)^2] = \frac{1}{2}ml^2[Δ\dot{θ}]^2 [/itex]
so [itex] Δ\dot{θ} = \frac{L}{l}\sqrt{[[\dot{θ(t_0)}]^2-(ω_0θ_0)^2]} [/itex]
Now in order to find the impulse [itex] Δp [/itex]: [itex] \|\vec{ΔL}\|=\|\vec{r}\|\|\vec{Δp}\|(1) = IΔ\dot{θ} → Δp = \frac{mL^2}{l}Δ\dot{θ}[/itex]
So finally I have [itex] Δp = \frac{mL^3}{l^2}\sqrt{[[\dot{θ(t_0)}]^2-(ω_0θ_0)^2]} [/itex]
I have a feeling I made a mistake in choosing my [itex]ΔE_+[/itex] any help would be greatly appreciated.
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