Fc is the centripetal force of the circular motion, m is the mass (kg) of the object undergoing the motion, v (m/s) is the linear or tangential velocity of the object and r (m) is the radius of this circular motion.
Centripetal force provides the curvature to an object’s circular motion. Its direction is orthogonal (right-angled) to the direction of the object’s motion (velocity). In other words, it is always towards the centre of the circular motion. Centripetal acceleration (which can be calculated by Newton’s second law) is in the same direction as centripetal force.
Centripetal force is a non-real force
This means that centripetal force always caused by a real force.
Relationship between centripetal force, mass, speed and radius
From the formula, we can deduce the following:
Effect on centripetal force
Changes in centripetal force
Concept Question 1
A car rounds a bend on a road that follows the arc of a circle with radius r. The car has mass m and is travelling at a velocity v. Explain the following situations:
(a) Why are drivers advised to slow down during wet weather, specifically when they are making a bend.
(b) Assuming the friction between the tyres and the road does not change, describe the path of a car with mass 2m when it rounds the bend at velocity v?
(c) A motor cyclist rounds the same bend at velocity 2v. If the mass of the motorcycle is 0.25m, what would be different about the centripetal force acting on the motorcycle compared to that on the car?
Concept Question 2
HSC Q30 2013
The diagram shows a futuristic space station designed to simulate gravity in a weightless environment.
Concept Questions Solutions
(a) During wet weather, the kinetic friction between a car's tyres and the ground is reduced. This means the centripetal force acting on the car during its bend is reduced. As a result, velocity needs to decrease to maintain radius of the curvature.
(b) Since centripetal force remains constant, the radius of curvature is doubled. This means for a car with mass 2m travelling at the same speed v, it requires a greater distance to complete the bend.
As seen above, substituting 0.25m and 2v into the equation `F_c=(mv^2)/r` will yield a magnitude of centripetal force identical to one with mass m and velocity v. This means the centripetal force acting on the motorcycle remains unchanged and so does its radius.
(a) The rotating motion of the spacecraft exerts normal force (centripetal force) on the astronaut. Due to Newton's third law, the astronaut exerts reaction force on the outer perimeter of the spacecraft. The acceleration resulted from this reaction force simulates gravity.
(b) As v increases, the magnitude of centripetal force increases (since radius remains constant). Changes in centripetal force are always proportional to the square of change in velocity.
(c) Yes, to maintain gravity, the magnitude of centripetal force cannot be changed. A decrease in radius needs to be compensated by a reduction in velocity v. This means the rotational speed to simulate gravity is lower for smaller space stations.
(d) No, the mass of the space station does not affect centripetal force nor acceleration. In addition, the mass of the astronaut does not influence the rotational speed required to achieve 1g of gravitational acceleration. This is because simulated gravity is independent of mass: `a_c=v^2/r`.
The key point that is missing from your two contradictory "explanations" is this:
While you are changing the radius, you are not moving in a circle around the original center point. You are moving in a spiral of some kind.
Think about a stone tied to a string being whirled in a circle. If you pull harder on the string to shorten it, the stone starts to spiral inwards, and the string is applying a tangential force to the stone as well as the radial force that causes the centripetal acceleration. The tangential force will increase the speed of the stone around the circle as the radius decreases.
So there are two changes which have opposite effects on the tension in the string. Reducing the radius and keeping everything else the same would reduce the centripetal acceleration, but increasing the speed and keeping everything else the same would increase it.
To find out which effect "wins" you need to do some math, but you said you don't want a "formula".
Also the question says "you" are moving in a circle, but it doesn't say how the centripetal force that keeps you moving in a circle is being applied to you.
So, you haven't fully described what the real-world system is, and you don't want to use the best way (math) to model how it behaves. Therefore, this isn't a complete answer!