A 9.5 kg monkey is hanging by one arm from a branch and is swinging on a vertical circle. As an approximation, assume a radial distance of 90 cm between the branch and the point where the monkey's mass is located. As the monkey swings through the lowest point on the circle, it has a speed of 3.4 m/s.
(a) Find the magnitude of the centripetal force acting on the monkey.
(b) Find the magnitude of the tension in the monkey's arm.
To find the magnitude of the centripetal force acting on the monkey and the magnitude of the tension in the monkey's arm, we can use the following steps:
(a) Find the magnitude of the centripetal force:
- The centripetal force is the net force acting towards the center of the circular motion.
- It can be calculated using the formula: Fc = m * v^2 / r, where m is the mass of the monkey, v is the speed, and r is the radial distance.
Let's calculate Fc:
Given:
Mass of the monkey (m) = 9.5 kg
Speed (v) = 3.4 m/s
Radial distance (r) = 90 cm = 0.9 m
Fc = (m * v^2) / r
= (9.5 kg) * (3.4 m/s)^2 / 0.9 m
≈ 44.97 N
Therefore, the magnitude of the centripetal force acting on the monkey is approximately 44.97 N.
(b) Find the magnitude of the tension in the monkey's arm:
- The tension in the monkey's arm is responsible for providing the centripetal force and is equal to the magnitude of the centripetal force.
- So, the tension in the monkey's arm is also approximately 44.97 N.
Therefore, the magnitude of the tension in the monkey's arm is approximately 44.97 N.
To find the answers to these questions, we need to use the concepts of centripetal force and tension.
(a) The centripetal force is the force that keeps an object moving in a circular path. It is always directed towards the center of the circle. In this case, the centripetal force is provided by the tension in the monkey's arm. We can use the formula for centripetal force:
F_c = (m * v^2) / r
where F_c is the centripetal force, m is the mass of the monkey, v is the speed of the monkey at the lowest point, and r is the radius of the circle.
Plugging in the given values:
m = 9.5 kg
v = 3.4 m/s
r = 0.9 m
F_c = (9.5 kg * (3.4 m/s)^2) / 0.9 m
F_c = 113.4 N
So, the magnitude of the centripetal force acting on the monkey is 113.4 N.
(b) To find the tension in the monkey's arm, we need to take into account the force of gravity acting on the monkey. At the lowest point of the circle, the tension in the arm must balance the gravitational force. The gravitational force is given by:
F_g = m * g
where F_g is the gravitational force and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the given value:
m = 9.5 kg
g = 9.8 m/s^2
F_g = 9.5 kg * 9.8 m/s^2
F_g = 93.1 N
Therefore, the magnitude of the tension in the monkey's arm is also 93.1 N since it must balance the gravitational force.
To summarize:
(a) The magnitude of the centripetal force acting on the monkey is 113.4 N.
(b) The magnitude of the tension in the monkey's arm is 93.1 N.