Chapter 5.3, Problem 19E

(a)

To determine

The solution of the second order linear differential equation $\frac{d^2\theta }{d{t}^2}+\left(\frac{g}{l}\right)\theta =0$ where, $\theta \left(t\right)$ is the angular displacement. The initial conditions are $\theta \left(0\right)=0\text{ and }{\theta }^{\prime }\left(0\right)={\omega }_0$.

(b)

To determine

To show: The relation between the velocity of the bullet $\left(v_b\right)$ and maximum angular displacement $\left({\theta }_{\max }\right)$ is $v_b=\left(\frac{m_w+m_b}{m_b}\right)\left(\sqrt{lg}\right){\theta }_{\max }$.

(c)

To determine

To show: The equation of the velocity of the bullet $\left(v_b\right)$ in terms of $m_w,m_b\text{ and }h$ is $v_b=\left(\frac{m_w+m_b}{m_b}\right)\sqrt{2gh}$.

(d)

To determine

The velocity of the bullet $\left(v_b\right)$ if the mass of the bullet is $5$ grams, mass of the wooden block is $1$ kilograms and the height to which the block is raised is $6$ centimeters.