Drive an expression to calculate the tension and velocity in the motion of a vertical Circle at the
1: highest point and 2: lowest point?
To calculate the tension and velocity at the highest and lowest points of a vertical circle, we can use the principles of circular motion and the concept of centripetal force.
1. Tension and velocity at the highest point:
At the highest point of the vertical circle, the tension in the string is at its minimum, and the velocity is also at its minimum. This is because the weight of the object provides the centripetal force required for circular motion.
To calculate the tension at the highest point, we need to consider the forces acting on the object at that point. These forces include the tension in the string (T) and the weight of the object (mg), where m is the mass of the object and g is the acceleration due to gravity.
Since the object is moving in a circular path, the net force acting on it should be the centripetal force (Fc) directed towards the center of the circle. At the highest point, the net force can be calculated as follows:
Fc = T - mg
To find the tension (T), we rearrange the equation:
T = Fc + mg
Next, to find the velocity at the highest point, we can use the concept of conservation of energy. At the highest point, the object has maximum potential energy and zero kinetic energy. The total mechanical energy (E) is conserved throughout the motion, so at the highest point:
E = mgh + 0.5mv^2
Where h is the height of the highest point and v is the velocity. At this point, all the energy is in the form of potential energy (mgh). Rearranging the equation, we can solve for the velocity (v):
v = sqrt(2gh)
2. Tension and velocity at the lowest point:
At the lowest point of the vertical circle, the tension in the string is at its maximum, and the velocity is also at its maximum. This is because, in addition to the weight of the object providing the centripetal force, the tension in the string helps provide the necessary force for circular motion.
To calculate the tension at the lowest point, we use the same equation as before:
T = Fc + mg
However, at the lowest point, the net force acting on the object is:
Fc = T - mg
To find the velocity at the lowest point, we again make use of the conservation of energy. At the lowest point, the object has maximum kinetic energy and zero potential energy. So the total mechanical energy (E) is:
E = 0 + 0.5mv^2
Simplifying the equation, we can solve for the velocity (v):
v = sqrt(2gh)
Note that the velocity at the highest and lowest points of the vertical circle is the same and depends only on the height (h) of those points. The only difference is the tension in the string, which is minimum at the highest point and maximum at the lowest point.