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A toy car, 50 g slides on a looping track as shown in the figure. The radius of the circular part of the track is 0.20 m.

a) We let the car start from point A, which is 0.60 m higher than B. Calculate the speed and normal force when the car passes B. Also calculate the normal force at point C.

enter image description here

My solution: From A to B Ep=>Ek mgh=mv²÷2 v≈3.4 m/s Fres In B: Fn-mg=mv²÷r =>Fn=3.4 N The total energy is conserved. In position C we have: E=mgh+mv²÷2 =>v in point C is 1.98m/s. The normal force at point C: Fn= mv²÷r -mg= 0.49 N. The problem is that Fc is not constant now. I know that I solved the question correctly but it doesn't make sense that the centripetal power is not constant. Help please

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  • $\begingroup$ Why do you expect a constant centripetal force? Wouldn't it make sense that the car slows down while moving up the loop, and speeds up again while moving down the loop, implying a varying centripetal acceleration? $\endgroup$
    – Steeven
    Commented Jun 1 at 13:05
  • $\begingroup$ My teacher told me that a centripetal force is always the same. $\endgroup$
    – user404180
    Commented Jun 1 at 13:07
  • $\begingroup$ That must have been in some specifical case then. Have a look at the formula for centripetal acceleration: $$a_c=v^2/r.$$ A constant centripetal force causes a constant centripetal acceleration, but this acceleration is only constant in certain very specific scenarios. One such scenario is when both the the radius $r$ of the circular path and the tangential speed $v$ around this path are both constant (this is called uniform circular motion). But if eg the speed changes along the way, which it would expectedly do in your scenario with the loop, then the centripetal force is no longer constant. $\endgroup$
    – Steeven
    Commented Jun 1 at 14:16

2 Answers 2

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No, in this particular case, centripetal force is not constant. We know that we can define centripetal force as: $$F = \frac{mv^2}{R}$$ where

  1. $m$ is the mass of particle in kg (Which is a toy car in your case).
  2. $v$ is the velocity of the particle in m/s.
  3. $R$ is the radius of the circular path in meters.

Here, centripetal force acts orthogonally towards the center of the circular track.

Since the velocity of the toy car is variable, the centripetal force would also be variable which can be easily deduced by the aforementioned formula (i.e. since mass and radius are constants, plugging different values of velocity would yield different values for the centripetal force.)

Hence the magnitude of centripetal force will vary throughout the trajectory of the particle. Had it been a case of uniform motion, the magnitude of centripetal force would have been constant throughout the motion.

Hope you query is resolved.

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What do you mean by "it doesn't make sense"?

First of all, UNIFORM circular motion (not this case) is the only case where the centripetal force is always constant, and that too is only in magnitude.

In this case, the centripetal force may not be constant in magnitude but the normal contact force and gravitational force balance it at every instant of time so that the next centripetal force causes the body to move in a perfect circle.

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  • $\begingroup$ With 'it doesn't make sense' I meant that this focre thing became a paradox, since the force is always constant but not in such cases. However I understand now, thank you sm! $\endgroup$
    – user404180
    Commented Jun 1 at 13:15