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, we can use the equation for centripetal force:
Fc = m * v^2 / r
where Fc is the centripetal force, m is the mass of the monkey, v is the velocity of the monkey at the lowest point, and r is the radial distance between the branch and the monkey's mass.
(a) Finding the magnitude of the centripetal force:
Given:
m = 9.5 kg (mass of the monkey)
v = 3.4 m/s (velocity of the monkey)
r = 90 cm = 0.9 m (radial distance)
Plug in the values into the formula:
Fc = (9.5 kg) * (3.4 m/s)^2 / 0.9 m
= (9.5 kg) * (11.56 m^2/s^2) / 0.9 m
≈ 122.32 N
So, the magnitude of the centripetal force acting on the monkey is approximately 122.32 N.
(b) 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.
At the lowest point, there are two forces acting on the monkey:
1. The weight of the monkey acting downward (mg).
2. The tension in the monkey's arm acting upward (Tension).
Since the monkey is in equilibrium at the lowest point, the sum of these forces must be zero:
mg + Tension = Fc
Given:
m = 9.5 kg (mass of the monkey)
g = 9.8 m/s^2 (acceleration due to gravity)
Fc = 122.32 N (centripetal force)
Rearranging the equation above, we can solve for the tension:
Tension = Fc - mg
Plug in the values into the formula:
Tension = 122.32 N - (9.5 kg) * (9.8 m/s^2)
≈ 122.32 N - 93.1 N
≈ 29.22 N
So, the magnitude of the tension in the monkey's arm is approximately 29.22 N.
To solve this problem, we will need to use the principles of circular motion and centripetal force.
(a) The centripetal force is given by the equation:
Fc = (m * v^2) / r
Where:
Fc is the centripetal force
m is the mass of the object (monkey in this case)
v is the velocity
r is the radius of the circle
In this case, the mass of the monkey is given as 9.5 kg, and the velocity at the lowest point is given as 3.4 m/s. The radius of the circle is given as 90 cm, which is equivalent to 0.9 meters. Substituting these values into the equation, we have:
Fc = (9.5 kg * (3.4 m/s)^2) / 0.9 m
Calculating this expression, we find that the magnitude of the centripetal force acting on the monkey is approximately 139.70 N.
(b) The tension in the monkey's arm is what provides the centripetal force to keep the monkey moving in a circular path. At the lowest point of the swing, there are two forces acting on the monkey: the tension force in the arm and the force of gravity.
The equation for the net force at the lowest point is:
ΣF = Ft - mg
Where:
ΣF is the net force
Ft is the tension force
m is the mass of the monkey
g is the acceleration due to gravity (approximately 9.8 m/s^2)
At the lowest point of the swing, the monkey experiences the maximum force of gravity, which is given by:
Fg = m * g
The tension force in the arm needs to balance out the force of gravity, so Ft = Fg.
Therefore, the magnitude of the tension in the monkey's arm is also approximately 139.70 N.