A 2.5 kg object is whirled in a vertical circle whose radius is 0.89 m. If the time of one revolution is 0.94 s, the magnitude of the tension in the string (assuming uniform speed) when it is at the top is
To find the magnitude of the tension in the string at the top of the vertical circle, we need to consider the forces acting on the object. At the top of the circle, the object experiences two forces: the tension force from the string and the gravitational force.
First, let's calculate the gravitational force acting on the object. The gravitational force can be calculated using the formula:
F_gravity = m * g
where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s²).
Given that the mass of the object is 2.5 kg, the gravitational force can be calculated as:
F_gravity = 2.5 kg * 9.8 m/s² = 24.5 N
Next, let's consider the net force acting on the object at the top of the vertical circle. The net force is the difference between the tension force and the gravitational force.
Net force = Tension force - Gravitational force
Since the object is moving in a vertical circle at a constant speed, the net force is equal to the centripetal force:
Net force = Centripetal force
The centripetal force can be calculated using the formula:
Centripetal force = (mass * velocity²) / radius
where mass is the mass of the object, velocity is the velocity of the object, and radius is the radius of the circular path.
In this case, we are given the time for one revolution (0.94 s), and we can use this to calculate the velocity of the object. The velocity can be calculated as:
Velocity = (2 * π * radius) / time
Given that the radius is 0.89 m and the time is 0.94 s, the velocity can be calculated as:
Velocity = (2 * 3.14159 * 0.89 m) / 0.94 s ≈ 5.99 m/s
Now we can substitute the known values into the formula for the centripetal force:
Net force = (mass * velocity²) / radius
= (2.5 kg * (5.99 m/s)²) / 0.89 m
≈ 33.33 N
Since the net force is equal to the centripetal force, we can equate the net force to the tension force:
Tension force - Gravitational force = Net force
Tension force = Net force + Gravitational force
Tension force = 33.33 N + 24.5 N
Tension force ≈ 57.83 N
Therefore, the magnitude of the tension in the string when the object is at the top of the vertical circle is approximately 57.83 N.