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Problem 3 (35%) A heavy particle is initially placed at the top of a fixed vertical hoop of a radius of R. The particle is then struck by an impulsive horizontal blow, which introduces an initial angular velocity $\omega_0$ (or equivalently an initial horizontal velocity $v_0 = \omega_0 R$). Calculate the reaction of the hoop on the particle by means of the Lagrange's undetermined multipliers method. Specifically, (a) Write down the Lagrangian for the multiplier method. (b) Find $\ddot{\theta}(\theta)$, $\dot{\theta}^2(\theta, \omega_0)$, and the constraint force $N(\theta, \omega_0)$. (c) Find the angle $\theta_c$ at which the particle falls off as a function of $\omega_0$. (d) Find the critical value $\omega_0$, beyond which the particle would immediately fly off the hoop after the blow, i.e., at $\theta = 0$.

          Problem 3 (35%) A heavy particle is initially placed at the top of a fixed vertical hoop of a radius of R. The particle is then struck by an impulsive horizontal blow, which introduces an initial angular velocity $\omega_0$ (or equivalently an initial horizontal velocity $v_0 = \omega_0 R$). Calculate the reaction of the hoop on the particle by means of the Lagrange's undetermined multipliers method. Specifically, (a) Write down the Lagrangian for the multiplier method. (b) Find $\ddot{\theta}(\theta)$, $\dot{\theta}^2(\theta, \omega_0)$, and the constraint force $N(\theta, \omega_0)$. (c) Find the angle $\theta_c$ at which the particle falls off as a function of $\omega_0$. (d) Find the critical value $\omega_0$, beyond which the particle would immediately fly off the hoop after the blow, i.e., at $\theta = 0$.

$Problem 3 (35%) A heavy particle is initially placed at the top of a fixed vertical hoop of a radius of R. The particle is then struck by an impulsive horizontal blow, which introduces an initial angular velocity $\omega0$ (or equivalently an initial horizontal velocity $v0 = \omega0 R$). Calculate the reaction of the hoop on the particle by means of the Lagrange's undetermined multipliers method. Specifically, (a) Write down the Lagrangian for the multiplier method. (b) Find $\ddot{\theta}(\theta)$, $\dot{\theta}^2(\theta, \omega0)$, and the constraint force $N(\theta, \omega0)$. (c) Find the angle $\thetac$ at which the particle falls off as a function of $\omega0$. (d) Find the critical value $\omega0$, beyond which the particle would immediately fly off the hoop after the blow, i.e., at $\theta = 0$.$

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