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When we speak of a body moving in circular motion, we assume it to be moving in a horizontal circle. However, the body can also have movement in vertical fashion. Concepts like angular velocity, centripetal force, and centripetal acceleration come into play while we try to understand these concepts.
Acceleration is nothing but the change in velocity with respect to time. Since velocity is a vector quantity, it can change in two ways: either the direction of the body changes or the magnitude of its velocity. When the concept of velocity is talked of in circular motion, it keeps constantly changing direction-wise, even if its magnitude remains constant. This is because the velocity is tangential.
Suppose the radius of the circular path is r, then its acceleration = v2/r.
The distance travelled by the object is equal to the circumference of the circle, which is equal to 2𝞹r.
However, the displacement is zero because the starting and ending point of the body is the same.
Motion of an Object in a Vertical Plane
Consider you have an eraser of mass m, and you tie that to a thread and start rotating it in a vertical circle. The radius of that circle is, say, r. Now, when you do so, the acceleration of the eraser changes. It increases when the body goes downwards to the vertical circle and decreases when the eraser goes upwards to the vertical circle. Due to this, the motion is not uniform circular motion. Also, irrespective of the position of the eraser, its mass and gravity acting upon it ‘mg’ always act downwards.
Now, consider there is point p, where the velocity of the eraser is v, and the lowest point of the vertical circle is taken as l. Also, the distance between point p and point l is supposed to be d, and the velocity of the eraser at l is supposed to be u.
Applying the law of conservation of energy, energy at any two points in the circle will remain the same. Therefore,
Energy at p = Energy at l;
12mv2 + mgd = 12mu2
v2+2gd=u2
v2=u2-2gd
v=√u2-2gd
This v calculated is the velocity of the eraser at any point in that vertical circle.
Now, let’s calculate the tension created in the thread used. Assume that the centripetal force acts on point p; then, we can write
T-mg cosθ = mv2r
T = mv2r + mg cosθ
Cos θ = r-dr
Substitute this θ to calculate the value for T
T = mv2r + mg(r-dr)
T= mr (g(r-d)+v2) [Since, v2=u2-2gd]
T= mr (gr-gd+u2-2gd)
T= mr [u2-3gd+gr]
Therefore, this value of T tells how much tension gets developed in the string.
Motion of an Object in a Horizontal Path
When the eraser in the above example moves in a horizontal path, it experiences a force on itself called the centripetal force, which is always experienced in the direction towards the centre of the circle.
It is essential to understand that the centripetal force acting on the eraser is no new force; it is the summation of all the forces acting on it towards the centre of the circle. It is also essential that any force acting on the body in a vertical direction should be balanced, so that net vertical force is equal to 0 Newton (N). If not, the body will start accelerating in a vertical direction.
Period and frequency when talking of horizontal force are as follows:
T = total time/number of revolutions
But f = 1/T
So, f = number of revolutions/total time,
where time is measured in seconds and frequency in Hertz
Centripetal acceleration in horizontal motion
ac = v2r
ac = 4π2rT2
Conclusion
Motion in a circular path depends on various factors and works on the energy conservation theorem. Velocity changes but the distance from a point remains the same. When we say velocity is changing, we mean that the direction component of velocity is changing and not magnitude. This change in acceleration is also termed centripetal acceleration.
Various other examples of these types of forces can be seen, for instance, when you fill a bucket with water, tie it with a rope, and start rotating the bucket. You will witness that the water in it doesn’t fall. This is an application of circular motion in a vertical plane.
