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.

(a) The centripetal force acting on the monkey can be found using the centripetal force formula:

Fc = mv^2 / r

Where Fc is the centripetal force, m is the mass of the monkey, v is the velocity, and r is the radius.

Plugging in the values given, we have:

Fc = (9.5 kg)(3.4 m/s)^2 / (0.9 m)

Calculating this, we find:

Fc = 141.02 N

So, the magnitude of the centripetal force acting on the monkey is approximately 141.02 N.

(b) The tension in the monkey's arm can be found using the tension formula:

T = Fc + mg

Where T is the tension, Fc is the centripetal force, m is the mass of the monkey, and g is the acceleration due to gravity.

Plugging in the values given, we have:

T = 141.02 N + (9.5 kg)(9.8 m/s^2)

Calculating this, we find:

T ≈ 232.9 N

So, the magnitude of the tension in the monkey's arm is approximately 232.9 N.

To find the magnitude of the centripetal force acting on the monkey, we can use the formula for centripetal force:

F_c = m * a_c

where F_c is the centripetal force, m is the mass of the monkey, and a_c is the centripetal acceleration.

The centripetal acceleration can be calculated using the formula:

a_c = v^2 / r

where v is the velocity of the monkey and r is the radius of the circular path.

Given:
m = 9.5 kg
v = 3.4 m/s
r = 0.90 m

(a) Finding the magnitude of the centripetal force:

First, calculate the centripetal acceleration:
a_c = (3.4 m/s)^2 / 0.90 m = 12.74 m/s^2

Then, substitute the values into the centripetal force formula:
F_c = (9.5 kg) * (12.74 m/s^2) = 121.03 N

Therefore, the magnitude of the centripetal force acting on the monkey is 121.03 N.

(b) Finding the magnitude of the tension in the monkey's arm:

The tension in the monkey's arm provides the centripetal force, so the magnitude of the tension is equal to the magnitude of the centripetal force.

Therefore, the magnitude of the tension in the monkey's arm is also 121.03 N.

To find the magnitude of the centripetal force acting on the monkey, we can use the following equation:

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 on the circle, and
r is the radius of the circle.

Given:
m = 9.5 kg
v = 3.4 m/s
r = 90 cm = 0.9 m

(a) Plugging the given values into the equation, we can calculate the centripetal force:

F_c = (9.5 kg) * (3.4 m/s)^2 / (0.9 m)
F_c ≈ 127.16 N

Therefore, the magnitude of the centripetal force acting on the monkey is approximately 127.16 N.

To find the magnitude of the tension in the monkey's arm, we need to consider the forces acting on the monkey at the lowest point on the circle. The tension in the monkey's arm provides the necessary centripetal force to keep it moving in the circular path.

At the lowest point, the monkey's weight (m*g) acts downward, while the tension (T) acts upward. The net force in the vertical direction is F_net = m*g - T.

At the lowest point, the monkey is momentarily weightless (i.e., the normal force cancels out its weight), so the net force is solely provided by the tension T.

(b) Since F_net = T, the magnitude of the tension in the monkey's arm is equal to the magnitude of the net force:

T = F_net = m*g

Plugging in the given values:

T = (9.5 kg) * (9.8 m/s^2) ≈ 93.1 N

Therefore, the magnitude of the tension in the monkey's arm is approximately 93.1 N.