Problem 25 A monkey is swinging from tree t... [FREE SOLUTION] (2024)

Short Answer

Circular Motion

The monkey swinging on the vine is an example of circular motion. This means the monkey is moving along a circular path. For an object to move in a circular path instead of a straight line, there must be a force that constantly pulls it towards the center of the circle. This force is called centripetal force. When the monkey is at the bottom of the swing, it is at the lowest point of its circular motion. The critical aspect here is that, regardless of where the monkey is in the swing, there must always be a force pulling it toward the center of the circle. This keeps the monkey moving in a curve rather than straight.

Gravitational Force

Gravitational force is one of the key forces acting on the monkey. Gravity pulls the monkey downward towards the center of the Earth. We can calculate the gravitational force using the formula:
\( F_g = mg \)
Here, \( F_g \) is the gravitational force, \( m \) is the mass of the monkey, and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²). At the bottom of the swing, gravity continues to pull the monkey downward, which will be an essential factor in determining the net force that acts as the centripetal force.

Tension in the Vine

The tension in the vine contributes significantly to the net force acting on the monkey. Tension is the force exerted by the vine on the monkey, pulling it towards the vine’s point of attachment to the tree. If we denote the tension force as \( T \), this force acts along the direction of the vine. When the monkey is at the bottom of the swing, the tension force partly counteracts gravity’s downward pull. To keep moving in a circular path, the net inward force (towards the circle’s center) must be sufficient. As we will see, the tension in the vine plays a critical role in this net centripetal force.

Net Force Calculation

To find the centripetal force at the bottom of the swing, we need to look at the net force resulting from the forces acting on the monkey. Gravity (\( mg \)) and the tension in the vine (\( T \)) together determine the net force. The centripetal force (\( F_c \)) required to keep the monkey in its circular path can be calculated using:
\( F_c = T - mg \)
In this equation, \( mg \) (gravitational force) is subtracted from the vine’s tension force because gravity pulls down, while the tension partially counteracts this downward pull. The remaining force is what directs the monkey inward towards the path's center, thus serving as the centripetal force. This net force ensures that the monkey keeps swinging in a curved trajectory.