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We say an object experienced uniform circular motion iff it moves in a circle of constant radius, with constant angular velocity.

To experience uniform circular motion, must the net force on the object be directed towards the center of the circle? For example, a person in a graviton ride experiences the normal force on his back, which is directed towards the center of the graviton, and also the static friction force which allows his seat to move him in a circle. These forces are perpendicular, with the normal force acting radially and the static friction force acting tangentially. But the person still meets all the criteria for uniform circular motion, right?

Secondly, if an object undergoes uniform circular motion, what can be said about the centripetal force? Is the centripetal force equal to the net force on the object, or is it equal to the net force in the radial direction. I believe it is the latter, since if we are to calculate the centripetal force on the person in the graviton, we should not need to worry about the tangential static friction force. But, according to the derivation of the acceleration of a body undergoing circular motion found here the net acceleration, and hence the net force, acts radially.

Thirdly, if the motion is non-uniform, can we say that the net radial force is equal to the standard centripetal acceleration, $v^2/r$?

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  • $\begingroup$ 'net force' causes acceleration. In case of uniform circular motion, net force causes acceleration towards the center of circle, which keeps the motion circular. Such force is called centripetal force $\endgroup$
    – KawaiKx
    Jan 25, 2015 at 0:13
  • $\begingroup$ some force is also required to counter the static friction force. But that would be balancing in nature. no net effect! so no acceleration. Newton's first law - no net force, no acceleration, constant state (either constant position or constant motion in a straight line) $\endgroup$
    – KawaiKx
    Jan 25, 2015 at 0:17

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  1. If the motion is uniform (no angular acceleration) you do not need any tangential force, at least ideally. It's during the acceleration that you feel the force throwing you out of the chair. Also related: When does centripetal force cause constant circular motion?.

  2. I'm not sure I understand your second question. The centripetal force is the net force on the object and is in the radial direction. What's the problem with that? A (maybe?) related question is: Centripetal force and circular motion

  3. From the same wikipedia page you linked: As a generalization of the uniform circular motion case, suppose the angular rate of rotation is not constant. The acceleration now has a tangential component. The radial component of the acceleration is still given by $$ a_{\textbf{r}}(t) = - \frac{v^2(t)}{r} = - \omega^2(t) r $$ but now you have that $\omega=\omega(t)$ changes over time, so $a_{\textbf{r}}$ while still given by the same expression, gets bigger (in case the angular velocity is incresing) to be able to mantain the object on the same trajectory, opposing the increasing centrifugal force. The total acceleration is thus $$ \textbf{a}(t) = a_{\textbf{r}}(t) \hat{r}(t) + a_\theta(t) \hat{\theta}(t) $$ where $\hat{r}(t)$ and $\hat{\theta}(t)$ are the unit vectors respectively in the radial and tangential directions. The tangential component $a_\theta$ is needed to actually change the angular velocity.

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  • $\begingroup$ Thanks for the answer, but it does not answer the main issue pointed out in my post. There are situations, like the graviton example, when there is a radial force AND a tangential force, but you still move in a circle of constant radius with constant angular velocity. So I am asking, in these situations, is the centripetal force equal to the net force or to the net force in the radial direction? $\endgroup$
    – math_lover
    Dec 3, 2014 at 22:10
  • $\begingroup$ if there is a radial and a tangential force and nothing else, you don't move in a uniform circolar motion. As I said above, the tangential "friction" force is not actually needed to mantain you in the orbit. $\endgroup$
    – glS
    Dec 4, 2014 at 7:25
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1- yes for a body to experience uniform circular motion the net force must be directed towards center of the the circle. In a gravitron ride the walls are slanted. Therefore normal acts at an angle with the horizontal. The vertical component of normal balances gravitational force while the horizontal component provides the necessary centripetal force to the body. Lets say that the graviton follows uniform circular motion, that is it has a constant angular velocity. At that moment angular velocity of every person in the graviton is also same. In this case no friction acts, thus completing the conditions for uniform circular motion. The Moment that person tries to move, his angular velocity changes, thus friction acts to prevent that change. In other words friction will only acts when the person tries to change his value of angular velocity, or only in cases of non uniform circular motion. Angular velocity must be constant for uniform circular motion. Clearly from this we can say that in a condition of uniform cirular motion net force must act radially. There can tangential forces, but those forces must be balanced as net acceleration on the object is radial, thus net force must also be radial.

2- it is the former, explained above.

3- yes you are right when you say that. Just note that v is the instantaneous velocity of that body at that moment. In non uniform circular motion centripetal force constantly changes to equal itself to v^2/r to cause the body to remain in a circle.

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