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.
centripetal force= mv^2/r
tension= mg+mv^2/r
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To solve this problem, we need to use the concept of centripetal force.
(a) Find the magnitude of the centripetal force acting on the monkey.
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 monkey, v is its velocity, and r is the radius of the circular path.
Given:
m = 9.5 kg (mass of the monkey)
v = 3.4 m/s (velocity of the monkey)
r = 90 cm = 0.9 m (radius of the circular path)
Plugging in the values into the equation:
Fc = (9.5 kg * (3.4 m/s)^2) / 0.9 m
Fc = (9.5 kg * 11.56 m^2/s^2) / 0.9 m
Fc = 105.82 N
Therefore, the magnitude of the centripetal force acting on the monkey is 105.82 N.
(b) Find the magnitude of the tension in the monkey's arm.
At the lowest point of the circle, the tension in the monkey's arm provides the necessary centripetal force to keep the monkey in circular motion.
The tension in the arm can be calculated using the equation:
T = Fc + mg
where T is the tension in the arm, Fc is the centripetal force, m is the mass of the monkey, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given:
Fc = 105.82 N (calculated earlier)
m = 9.5 kg (mass of the monkey)
g = 9.8 m/s^2 (acceleration due to gravity)
Plugging in the values into the equation:
T = 105.82 N + (9.5 kg * 9.8 m/s^2)
T = 105.82 N + 93.1 N
T = 198.92 N
Therefore, the magnitude of the tension in the monkey's arm is 198.92 N.
To solve this problem, we can use the equations for centripetal force and tension.
(a) The centripetal force acting on the monkey is given by the equation:
Fc = m * v^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 given values:
m = 9.5 kg
v = 3.4 m/s
r = 0.9 m (converted from 90 cm to meters)
Fc = (9.5 kg) * (3.4 m/s)^2 / 0.9 m
Calculating this expression, the magnitude of the centripetal force acting on the monkey is approximately 137.15 N.
(b) The tension in the monkey's arm is equal to the centripetal force acting on the monkey at the lowest point of the swing.
So, the tension (T) is also equal to 137.15 N.
To summarize:
(a) The magnitude of the centripetal force acting on the monkey is approximately 137.15 N.
(b) The magnitude of the tension in the monkey's arm is approximately 137.15 N.