Chapter 7, Problem 32P

To determine

The horizontal and vertical components of the pendulum’s displacement.

Answer to Problem 32P

The vertical component of the pendulum displacement is $0.16\text{ m}$

The horizontal component of the pendulum displacement is $0.94\text{ m}$

Explanation of Solution

Given:

Mass of the rifle bullet is $m=28\text{ g = 28}\times {\text{10}}^{-3}\text{ kg}$

Velocity of the bullet is $v=230\text{ m/s}$

Mass of the pendulum is $\text{M = 3}\text{.6 kg}$

Length of the pendulum string is $L=2.8\text{ m}$

Formula used:

The pendulum is pictorially represented as shown in the figure 1.

  

Figure 1

The vertical component of the pendulum displacement can be calculated using the equation,

  $v=\frac{m+M}{M}\sqrt{2gh}$

  $\Rightarrow h=\frac{1}{2g}{\left(\frac{mv}{m+M}\right)}^2$

The horizontal component of the pendulum displacement can be calculated using the equation,

  $L^2={\left(L-h\right)}^2+x^2$

  $\Rightarrow x=\sqrt{L^2-{\left(L-h\right)}^2}$$\left(2\right)$

Calculation:

Substituting the numerical values in equation $\left(1\right)$ and $\left(2\right)$

  $h=\frac{1}{2\left(9.8{\text{ m/s}}^2\right)}{\left(\frac{\left(\text{28}\times {\text{10}}^{-3}\text{ kg}\right)230\text{ m/s}}{\text{28}\times {\text{10}}^{-3}\text{ kg}+\text{3}\text{.6 kg}}\right)}^2=0.16\text{ m}$

  $x=\sqrt{{\left(2.8\text{ m}\right)}^2-{\left(2.8-0.16\right)}^2}=0.94\text{ m}$

Conclusion:

The vertical component of the pendulum displacement is $0.16\text{ m}$

The horizontal component of the pendulum displacement is $0.94\text{ m}$