4. You have landed your spaceship on an asteriod and want to determine the acceleration due to gravity using a simple pendulum of 0.75 m If the period of this pendulum is 2.55, what is the value of g on the asteriod?

To determine the acceleration due to gravity on the asteroid using the simple pendulum, we can use the formula for the period of a simple pendulum:<br /><br />\[ T = 2\pi \sqrt{\frac{L}{g}} \]<br /><br />Where:<br />- \( T \) is the period of the pendulum<br />- \( L \) is the length of the pendulum<br />- \( g \) is the acceleration due to gravity<br /><br />Given:<br />- \( T = 2.55 \) seconds<br />- \( L = 0.75 \) meters<br /><br />We need to solve for \( g \):<br /><br />1. Rearrange the formula to solve for \( g \):<br /><br />\[ g = \frac{4\pi^2 L}{T^2} \]<br /><br />2. Substitute the given values into the formula:<br /><br />\[ g = \frac{4\pi^2 \times 0.75}{(2.55)^2} \]<br /><br />3. Calculate the value:<br /><br />\[ g = \frac{4 \times 9.8696 \times 0.75}{6.5025} \]<br /><br />\[ g = \frac{29.6072}{6.5025} \]<br /><br />\[ g \approx 4.57 \, \text{m/s}^2 \]<br /><br />Therefore, the acceleration due to gravity on the asteroid is approximately \( 4.57 \, \text{m/s}^2 \).