Chapter 6, Problem 29CT

Hungarian Adrian Annus won the gold medal for the hummer throw at the $2004$ Olympics in Athens with a winning distance of $83.19$ meters*.The event consists of swinging a $16$-pound weight attached to a wire $190$ centimetres long in a circle and then releasing it. Assuming his release is at a ${45}^{\circ }$ angle to the ground, the hammer will travel a distance of $\frac{v_0{}^2}{g}$ metres, where $g=9.8$ metres/second ${}^2$ and $v_0$ is the linear speed of the hammer when released. At what rate (rpm) was he swinging the hammer upon release?

*Annus was stripped of his medal refusing to cooperate with postmedal drug testing.

To determine

The rate (rpm) that Annus swinging the hammer upon release, if Hungarian Adrian Annus won the gold medal for the hammer throw at the $2004$ Olympics in Athens with a winning distance of $83.19$ meters. The event consists of swinging a $16$ pound weight attached to a wire $190$ centimeters long in a circle and release it. Assuming his release is at ${45}^0$ angle to the ground, the hammer will travel a distance of $=\frac{v_0{}^2}{g}$ meters where $g=9.8m/{\sec }^2$ and $v_0$ is the linear speed of the hammer when released.

Answer to Problem 29CT

Solution:

The rate (rpm) that Annus swinging the hammer upon release is $143.5\text{rev/m}$

Explanation of Solution

Given information:

Hungarian Adrian Annus won the gold medal for the hammer throw at the $2004$ Olympics in Athens with a winning distance of $83.19$ meters. The event consists of swinging a $16$ pound weight attached to a wire $190$ centimeters long in a circle and release it. Assuming his release is at ${45}^0$ angle to the ground, the hammer will travel a distance of $=\frac{v_0{}^2}{g}$ meters, where $g=9.8m/{\sec }^2$ and $v_0$ is the linear speed of the hammer when released.

Explanation:

To find the rate (rpm) that Annus swinging the hammer upon the release, that is the angular speed of hammer, first want to find the linear speed $v_0$

The distance travelled by the hammer is $\frac{v_0{}^2}{g}$ meters

The winning distance is $83.19$ meters $g=9.8m/{\sec }^2$

Therefore,

$\frac{v_0{}^2}{9.8}=83.19$

$\Rightarrow v_0{}^2=83.19\times 9.8=815.262$

$\Rightarrow v_0=\sqrt{815.262}=28.553$ m/s

Therefore, the linear speed $v_0=28.553$ m/s

Use the formula

$\text{Angular speed}\text{=}\frac{\text{linear}\text{speed}}{\text{radius}}$

The event consists of swinging a $16$ pound weight attached to a wire $190$ centimeters long in a circle

Therefore, the $\text{radius}\text{=190}\text{cm}\text{=1}\text{.9m}$

Thus, $\text{Angular speed}\text{=}\frac{{\text{v}}_0}{\text{1}\text{.9}}\text{rad/s}$

$\Rightarrow \text{Angular speed}\text{=}\frac{28.553}{\text{1}\text{.9}}\text{=}\text{15}\text{.028}\text{rad/s}$

$\Rightarrow \text{Angular speed}\text{=15}\text{.208}\frac{\text{radians}}{\sec }\cdot \frac{\text{60 sec}}{\text{1 min}}\cdot \frac{1\text{revolution}}{2\pi \text{radians}}=143.5\text{rev/m}$

Thus, the rate (rpm) that Annus swinging the hammer upon release is $143.5\text{rev/m}$