A small sphere of mass 3 kg is attached to the end of a cord of length 2m and rotated in a vertical circle about a fixed point O. Determine the tension in the cord at the uppermost point of the trajectory if the speed at this point is 8m/s. Use g=9.8 m/s2.
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To determine the tension in the cord at the uppermost point of the trajectory, we can analyze the forces acting on the sphere at that point.
The forces acting on the sphere at the uppermost point are its weight (mg) and the tension in the cord (T). The tension in the cord provides the centripetal force required to keep the sphere moving in a circle.
At the uppermost point, the speed of the sphere is along the tangent to the circle. The net force acting on the sphere in the vertical direction (upwards) should be equal to zero in order to prevent the sphere from moving away from the circle.
Let's break down the forces acting on the sphere at the uppermost point:
1. Weight (mg):
The weight of the sphere is always acting downwards and can be calculated by multiplying its mass (m) with the acceleration due to gravity (g).
Weight (mg) = 3 kg × 9.8 m/s^2 = 29.4 N
2. Tension (T):
The tension in the cord provides the centripetal force required to keep the sphere moving in a circle. At the uppermost point, the tension should be equal to the sum of the weight and the centripetal force.
Centripetal force (Fc) = (m × v^2) / r
where:
m = mass of the sphere = 3 kg
v = speed of the sphere = 8 m/s
r = radius of the circle = length of cord = 2 m
Plugging in the values:
Centripetal force (Fc) = (3 kg × 8 m/s)^2 / 2 m = 96 N
Therefore, the tension in the cord at the uppermost point of the trajectory is:
T = Weight (mg) + Centripetal force (Fc) = 29.4 N + 96 N = 125.4 N
Hence, the tension in the cord at the uppermost point of the trajectory is 125.4 N.
To determine the tension in the cord at the uppermost point of the trajectory, we need to analyze the forces acting on the sphere.
At the uppermost point, the sphere is momentarily at rest for an instant (its velocity is zero) before it starts moving downward. Thus, the net force acting on the sphere at this point is directed toward the center of the circle (point O).
We can break down the forces acting on the sphere into two components: the gravitational force (mg) and the tension in the cord (T). Let's denote the angle between the cord and the vertical line as θ.
At the uppermost point, the tension in the cord is equal to the centripetal force required to keep the sphere moving in a circular path:
T = (m * v²) / r
Where:
T = Tension in the cord
m = Mass of the sphere (3 kg)
v = Speed of the sphere at the uppermost point (8 m/s)
r = Length of the cord (2 m)
Now, since the tension is acting at an angle with respect to the vertical, we can use trigonometry to find its vertical and horizontal components.
The vertical component of the tension (T_v) will be equal to the gravitational force (mg) minus the centripetal force:
T_v = mg - T
The horizontal component of the tension (T_h) is zero since there is no horizontal acceleration at the uppermost point.
Now, let's calculate the tension in the cord using the derived formulas:
T = (m * v²) / r
= (3 kg * (8 m/s)²) / 2 m
= 96 N
Therefore, the tension in the cord at the uppermost point of the trajectory is 96 N.