Harmonic Oscillations with Escapement [Clock]

[hkEkers]

Homework Statement

Clock activates escapement every time it passes through the vertical. Escapment under tension from a hanging weight that gives an impulse distance l from the pivot. Energy transferred by this compensates for the energy dissipation due to friction so the amplitude is constant.

(a)What is the impulse needed to sustain the motion of a pendulum length L mass m amplitude of swing theta_0 and quality factor Q

(b) Why is it desirable for the pendulum to engage the escapement as as it passes vertical rather than another point.

Homework Equations

Q=(m*omega)/F_dampening
omega= sqrt(k/m)

E(t) = 1/2 A^2 [ m*omega^2*sin^2(omega*t+phi) + k*cos^2(omega*t+phi)]
E(t) = K(t) + U(t)

The Attempt at a Solution

For a, either measuring energy loss by U(x,t)= U(x,t+T) + Energy loss which I'm not sure how to solve for without it being obnoxiously long and neglecting a couple variables. No idea how to account for distance from the pivotal point in the equation without involving Torque, and considering this is a module on harmonic oscillation seems less likely.

Well for b my guess would be it's when acceleration =0 so there aren't any other forces acting on the swing at the time or so the impulse isn't applied at an angle doesn't add an acceleration to multiple axes.

I'm completely lost on how to create a relationship between a lot of the elements in this problem. Any help would be appreciated.

 

[anathema]

Dear forum post author,

Thank you for sharing your question with us. I would like to offer some insights and suggestions to help you solve this problem.

Firstly, let's define some variables to make the problem more manageable. Let L be the length of the pendulum, m be the mass of the pendulum, theta_0 be the amplitude of swing, and Q be the quality factor.

(a) To find the impulse needed to sustain the motion of the pendulum, we can start by looking at the energy equations you have provided. We can express the total energy of the pendulum as the sum of kinetic energy (K) and potential energy (U). We know that the energy transferred by the escapement compensates for the energy dissipation due to friction, so we can equate K and U at any given time t. This gives us the equation E(t) = K(t) + U(t).

Now, we can use the equations for K and U to substitute in terms of the variables we defined earlier. This will give us E(t) = 1/2 m v^2 + 1/2 k x^2, where v is the velocity of the pendulum and x is the displacement from equilibrium.

Next, we can use the equation for the velocity of a pendulum, v = L*omega*sin(theta), where omega is the angular velocity and theta is the angle of displacement. We can also use the equation for angular velocity, omega = sqrt(k/m), to substitute for omega. This will give us v = sqrt(k/m)*L*sin(theta).

Now, we can substitute this value for v in our energy equation, giving us E(t) = 1/2 m * (sqrt(k/m)*L*sin(theta))^2 + 1/2 k * x^2.

We can simplify this equation to E(t) = 1/2 (kL^2*sin^2(theta) + kx^2).

Since we know that the amplitude is constant, we can set x = theta_0 and solve for the impulse needed to sustain the motion. This will give us the equation I = sqrt(2m*k)*theta_0, where I is the impulse.

(b) It is desirable for the pendulum to engage the escapement as it passes through the vertical because it maximizes the energy transfer from the hanging weight to the pendulum. At