Centripetal Force and Curving Motion

Created by Chinmay Kulkarni

Main Idea

A body having circular motion always has the centripetal force pointing inside the circle and the velocity tangential to the circle

A centripetal force is a force acting on a body while it has curving motion. In these certain situations, the momentum of the system is not constant, since the direction of motion, or velocity always changes direction while the speed may remain constant. For example, image you are kicking a ball that is moving perpendicular to the direction of motion. Now if you keep kicking the ball in this similar manner for a small duration of time, the ball would move in a circular path.

When an object is moving in vertical centripetal motion, for example when a man is swinging on a rope, the force gravity [math]\displaystyle{ mg }[/math] must be accounted with as well as the tension force [math]\displaystyle{ F_t }[/math].

Both the tension force as well as the gravitational force needs to be accounted.

Mathematical Model

While an object is in circular motion, the centripetal force is always perpendicular to the velocity and momentum of the object, meaning that the object experiences a force towards the centre of the circle while it is moving. The simple mathematical model for centripetal force is normally [math]\displaystyle{ F_c = ma_c = \frac{m v^2}{r} }[/math] for any object moving in a circle. However, since this is circular motion, many times the angular velocity ω in radians/second of the system moving is given.

In this case

[math]\displaystyle{ v = \omega r }[/math]

meaning

[math]\displaystyle{ F_c = {m\omega^2 r} }[/math]

However sometimes the period of rotation [math]\displaystyle{ T }[/math], in seconds is given

[math]\displaystyle{ T = \frac{2\pi}{\omega} \,. }[/math]

Thus, the equation becomes

[math]\displaystyle{ F = m r \left(\frac{2\pi}{T}\right)^2 }[/math]

However in many circumstances, it is helpful to split the centripetal force into parallel and perpendicular forces, or [math]\displaystyle{ F_{para} }[/math] and [math]\displaystyle{ F_{perp} }[/math] respectively.

This means

[math]\displaystyle{ F_c = F_{para} + F_{perp} }[/math]

Thus we can rewrite the momentum principle as follows

[math]\displaystyle{ Δp = (F_{para} + F_{perp})Δt }[/math]

Normally, [math]\displaystyle{ F_{para} }[/math] dictates the velocity of the object, while [math]\displaystyle{ F_{perp} }[/math] dictates the direction of motion.

Computational Model

A computational representation of centripetal force can be created using VPython.

We can find the momentum initial position, momentum, and forces in Python.

   while t <30000000:
   rate(100)
   scene.center=moon.pos/2
   scene.range=Earth.radius*60
   p_init=mag(pcraft)
   pcraft_i=pcraft+vector(0,0,0)                                                 #craft initial momentum
   r=Earth.pos-craft.pos                                                             #craft initial position from Earth
   rM=moon.pos-craft.pos                                                         #craft initial position from Mon
   Fgrav=G*mEarth*mcraft/(mag(craft.pos-Earth.pos)**2)     #gravity from earth
   FgravM=G*mcraft*mmoon/(mag(craft.pos-moon.pos)**2)      #gravity from moon
   pcraft=pcraft+r/mag(r)*Fgrav*deltat+rM/mag(rM)*FgravM*deltat    #new momentum of the craft based on Earth and Moon

From this we can calculate the change in momentum and the new positions based on the forces.

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The purple arrow represents the direction of force on a spacecraft orbiting earth. The yellow arrow represents the velocity of the spacecraft. The yellow line represents the motion of the craft.

   deltap=pcraft-pcraft_i      #calculates change in momentum
   Fnet=deltap/deltat          #calculates Fnet from Earth and Moon
   p_final=mag(pcraft)         
   Fnet_tangent=((p_final-p_init)/deltat)*norm(pcraft)     #calculates Fnet par.
   Fnet_tangent_arrow.pos=craft.pos
   Fnet_tangent_arrow.axis=Fnet_tangent*1000000
   Fnet_perp=Fnet-Fnet_tangent                                       #calculates Fnet perp
   Fnet_perp_arrow.pos=craft.pos
   Fnet_perp_arrow.axis=Fnet_perp*1000000
   vcraft=pcraft/mcraft                #calculates new velocity of the craft

Examples

Simple

An engineer whose mass is 75 kg holds onto the outer rim of a rotating space station whose radius is 16 m and which takes 45 s to make one complete rotation.

What is the magnitude of the force the engineer has to exert in order to hold on?

[math]\displaystyle{ F = m r \left(\frac{2\pi}{T}\right)^2 = 75* 16 \left(\frac{2\pi}{45}\right)^2 = 23.39 }[/math]

Middling

Tarzan whose mass is 90 kg is swinging on a vine. At the bottom most point the 8 m long vine breaks. At that point he is traveling at 12 m/s. What was the tension in the vine at the point it broke?

[math]\displaystyle{ \frac{mv^2}{r} = F_T - mg }[/math]

[math]\displaystyle{ \frac{90*12^2}{8} = F_T - 90*9.8 }[/math]

[math]\displaystyle{ F_T = 2502 N }[/math]

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