Chapter 7.10, Problem 44SEP

To determine

The raising height of the pendulum for achieving the fracture of the specimen.

Answer to Problem 44SEP

The raising height of the pendulum for achieving the fracture of the specimen is $0.305\text{m}$

Explanation of Solution

Write the expression of height of the hammer when it reached $120°$ from its initial position is,

    $h_1=l+l\sin 30°$                                                                            (I).

Here, raised height is $h_1$ and length of arm is $l$.

Write the expression for potential energy stored in the hammer is,

    $P.E_1=m\times g\times h_1$                                                                         (II).

Here, potential energy stored in the hammer is $P.E_1$,the, mass of the hammer is $m$, acceleration due to gravity is $g$ and the height of impact is $h_1$.

Write the expression of remaining energy after fracture of the specimen is,

    $E=P.E_1-E_1$                                                                        (III)

Here, remaining energy stored in the hammer is $E$, potential energy stored in the hammer is $P.E_1$ is and energy required for fracture is $E_1$.

Write the expression of energy used for lifting is,

    $\begin{array}{l}E=m\times g\times h_2\\ h_2=\frac{E}{m\times g}\end{array}$                                                                               (IV)

Here, mass of hammer is $m$, remaining energy is $E$, acceleration due to gravity is $g$ is and lifting height or impact height is $h_2$.

Conclusion:

Below figure represent the new position of the hammer when hammer is raised to $120°$,

Figure-(1)

From Figure-(1), when the hammer is rotated to an angle of $120°$, then the arm will become horizontal. From this height in which hammer raised $h_1$ height of length of the arm $l$ is found.

Substitute, $120\text{cm}$ for $l$ in Equation (I).

    $\begin{array}{c}h=\left(120\text{cm}\right)+\left(120\text{cm}\right)\sin 30°\\ =180\text{cm}\left(\frac{1\text{m}}{100\text{cm}}\right)\\ =1.8\text{m}\end{array}$

Substitute, $\text{15}\text{kg}$ for $m$, $9.81\text{m}/{\text{s}}^{\text{2}}$ for $g$ and height $1.8\text{m}$ for $h_1$ in Equation (II).

    $\begin{array}{c}P.E_1=15\left(\text{kg}\right)\times 9.81\left(\text{m}/{\text{s}}^{\text{2}}\right)\times 1.8\left(\text{m}\right)\\ =\left(246.87\text{kg}\cdot {\text{m}}^2/{\text{s}}^{\text{2}}\right)\left[\frac{1\text{N}}{1\text{kg}\cdot \text{m}/{\text{s}}^{\text{2}}}\right]\left[\frac{1\text{J}}{1\text{N}\cdot \text{m}}\right]\\ =\text{246}\text{.87}\text{J}\end{array}$

Substitute, $\text{246}\text{.87}\text{J}$ for $P.E_1$ and $220\text{J}$ for $E_1$ in Equation (III).

    $\begin{array}{c}E=264.87\text{J}-220\text{J}\\ =44.87\text{J}\end{array}$

This remaining energy is used for lifting of hammer.

Substitute, $44.87\text{Joule}$ for $E$, $15\text{kg}$ for $m$ and $9.81\text{m}/{\text{s}}^{\text{2}}$ for $g$ in Equation (IV).

    $\begin{array}{c}{h}_2=\frac{44.87\text{J}}{15\left(\text{kg}\right)\times 9.81\left(\text{m}/{\text{s}}^{\text{2}}\right)}\\ =\frac{\left(44.87\text{J}\right)\left[\frac{1\text{N}\cdot \text{m}}{1\text{J}}\right]\left[\frac{1\text{kg}\cdot \text{m}/{\text{s}}^{\text{2}}}{1\text{N}}\right]}{\left(15\text{kg}\cdot \text{m}/{\text{s}}^{\text{2}}\right)\times 9.81\left(\text{m}/{\text{s}}^{\text{2}}\right)}\\ =0.305\text{m}\end{array}$

Thus, the raising height of the pendulum for achieving the fracture of the specimen is $0.305\text{m}$