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(a) Substitute for r as a function of time, and derive the complete equation of motion for 0. (b) Introduce a new variable z such that, bz = a + bt. Rewrite the equation of motion in terms of z. (c) Derive the solution for 0(z)

(a) Substitute for r as a function of time, and derive the complete equation of motion for 0. (b) Introduce a new variable z such that, bz = a + bt. Rewrite the equation of motion in terms of z. (c) Derive the solution for 0(z)

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# Oscillations of a Pendulum Consider a simple pendulum in which a particle of mass \( m \), connected to a massless string, oscillates. As the pendulum oscillates, the length \( r \) of the string increases, such that at time \( t \) the length \( r \) is \( r = a + bt \), where \( a \) and \( b \) are two constants. The equation of motion for the angle \( \theta \) that the string makes with the vertical line (in the direction of gravity) is given by \[ \frac{d}{dt} \left( mr^2 \frac{d\theta}{dt} \right) + mrg \sin \theta = 0. \] We assume that the oscillations are small, so that \( \sin \theta \approx \theta \). 1. **Substitute for \( r \) as a function of time, and derive the complete equation of motion for \( \theta \).** 2. **Introduce a new variable \( z \) such that \( bz = a + bt \). Rewrite the equation of motion in terms of \( z \).** 3. **Derive the solution for \( \theta(z) \).** 4. **Determine the two constants of integration by using the initial conditions, \( d\theta/dt = 0 \) and \( \theta = \theta_0 (a/b)^{1/2} \) at \( t = 0 \).**
Transcribed Image Text:# Oscillations of a Pendulum Consider a simple pendulum in which a particle of mass \( m \), connected to a massless string, oscillates. As the pendulum oscillates, the length \( r \) of the string increases, such that at time \( t \) the length \( r \) is \( r = a + bt \), where \( a \) and \( b \) are two constants. The equation of motion for the angle \( \theta \) that the string makes with the vertical line (in the direction of gravity) is given by \[ \frac{d}{dt} \left( mr^2 \frac{d\theta}{dt} \right) + mrg \sin \theta = 0. \] We assume that the oscillations are small, so that \( \sin \theta \approx \theta \). 1. **Substitute for \( r \) as a function of time, and derive the complete equation of motion for \( \theta \).** 2. **Introduce a new variable \( z \) such that \( bz = a + bt \). Rewrite the equation of motion in terms of \( z \).** 3. **Derive the solution for \( \theta(z) \).** 4. **Determine the two constants of integration by using the initial conditions, \( d\theta/dt = 0 \) and \( \theta = \theta_0 (a/b)^{1/2} \) at \( t = 0 \).**
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