I just need the formula.
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 when the object is at the top of the vertical circle, you can use the following formula:
Tension = (mass × velocity^2) / radius + (mass × gravity)
Here's how you can derive this formula:
1. Start by recognizing that the object is undergoing uniform circular motion, meaning it has a constant speed throughout its motion but its direction changes continuously.
2. Let's first find the velocity of the object when it is at the top of the circle. We can use the distance traveled and the time taken for one revolution to find the linear speed:
Linear speed = distance / time
Given that the radius of the circle is 0.89 m and the time for one revolution is 0.94 s:
Linear speed = 2πr / time
= (2π × 0.89) / 0.94
3. Now, knowing the linear speed, we can calculate the velocity of the object at the top using the relationship between linear speed and velocity in circular motion:
Velocity = linear speed
4. The tension at the top of the circle must be sufficient to overcome two forces acting on the object: the inward centripetal force (providing the necessary acceleration towards the center) and the gravitational force pulling the object downwards.
The centripetal force is given by the equation:
Centripetal force = (mass × velocity^2) / radius
The gravitational force is given by the equation:
Gravitational force = mass × gravity
Where:
- mass is the mass of the object (given as 2.5 kg)
- velocity is the velocity of the object at the top
- radius is the radius of the circle (given as 0.89 m)
- gravity is the acceleration due to gravity (approximated as 9.8 m/s^2)
5. Finally, to find the tension at the top of the circle, add the two forces:
Tension = centripetal force + gravitational force
= (mass × velocity^2) / radius + (mass × gravity)
Substitute the given values into the formula to calculate the magnitude of the tension in the string.