Jeff of the Jungle swings on a vine that is r = 6.40 m long. At the bottom of the swing, just before hitting a tree, Jeff's linear speed is 7.10 m/s.
To find the answer to this question, we need to use the concept of centripetal force. The centripetal force is the force acting towards the center of a circular path that keeps an object moving in a circle. In this case, the centripetal force is provided by the tension in the vine.
The formula for centripetal force can be written as:
F = m * v² / r
Where:
- F is the centripetal force
- m is the mass of the object
- v is the linear velocity of the object
- r is the radius of the circular path
However, since we are only interested in the linear speed and radius, we can rearrange the formula to solve for mass:
m = F * r / v²
We are given the linear speed (v = 7.10 m/s) and the radius (r = 6.40 m). But we don't have the centripetal force (F). To solve for it, we need another equation involving the force.
Jeff is swinging right before hitting a tree, so the centripetal force is equal to the force of gravity (mg). Here, m is the mass of Jeff and g is the acceleration due to gravity (approximately 9.8 m/s²).
Now we can write the equation as:
mg = F
Substituting this value of F into the earlier equation and rearranging, we get:
m = (mg * r) / v²
Since we have the acceleration due to gravity (g = 9.8 m/s²), we can solve for mass:
m = (9.8 * 6.40) / (7.10²) = 5.836 kg
Therefore, Jeff's mass is approximately 5.836 kg.