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When analyzing centripetal motion, it comes to mind that the motion is caused by some central force that constantly pulls on a rotating body. This force naturally causes acceleration towards the center. I understand that the acceleration arrises from the change in velocity, namely, the direction of the velocity; however, I fail to see why the magnitude of the velocity stays constant. Even if the acceleration is at a different direction, should'nt it affect the velocities magnitude? For example, when an object is decelerated: the acceleration is directed opposite to the velocity; and this acceleration both causes the direction of the velocity to change until it coincides with the direction of the acceleration, and it also changes the magnitude of the velocity. Why does'nt centripetal acceleration change the magnitude of the velocity of a rotating object?

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  • $\begingroup$ If you push a rock horizontally off a cliff, will gravity speed up or slow down its horizontal speed? $\endgroup$
    – RC_23
    Mar 18, 2023 at 23:05
  • $\begingroup$ No, however, it will increase it's vertical velocity. Does this mean that the centripetal acceleration will increase the magnitude of the velocity vertically? However, after one quadrant of the circle has been traced, the acceleration will then start accelerating it horiztontally, and then alternating between increasing vertical velocity and horizontal velocity. Why is this? $\endgroup$
    – Camelot823
    Mar 18, 2023 at 23:18
  • $\begingroup$ Any force at 90 degrees to the motion of an object will only change its direction, not its speed. In a circular motion, the force is always perpendicular to the velocity because a radius is always perpendicular to the arc of a circle. In general, you can characterize any trajectory of motion by splitting the forces on it into two types: tangential and normal (radial). The tangential force will do work on the object and change its speed, and the normal will change its direction but not its speed and do no work. $\endgroup$
    – RC_23
    Mar 19, 2023 at 19:17
  • $\begingroup$ The gravity example was not the best, because once the object starts falling, gravity is no longer wholly perpendicular to the motion, and does increase the speed. $\endgroup$
    – RC_23
    Mar 19, 2023 at 19:19

4 Answers 4

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$$ d(||v||^2) = d(v^2) = d(\vec v \cdot \vec v) $$

$$ =\vec v \cdot \vec{dv} + \vec{dv}\cdot \vec v$$ $$ =2\vec v \cdot \vec{dv} $$ $$ = 2\vec v \cdot (\frac{d\vec v}{dt})dt $$ $$ = 2\vec v \cdot \vec a dt $$ $$ = \frac 2 m \vec v \cdot \vec F dt $$

and for a centripetal force:

$$ \vec F \cdot \vec v \equiv 0 $$

so

$$ d(||v||) = 0 $$

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Because the centripetal force is at right angles to the body's velocity, it does no work on the body, so does not change its kinetic energy.

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You can drive a car at a constant speed. When the road is straight, this is easy to understand.

If you drive around a bend, you can keep the speedometer at a constant value, though the direction keeps changing. This changing direction is a change in velocity. A change in velocity is an acceleration. You can feel the sideways force. Why does the friction act on the inward direction when a car makes a turn on a level road?

To make the car go faster, the engine pushes it forward, in the direction it is traveling. To make it go slower, the brakes hold it back, against the direction of travel. If the force has no forward or backward component, it will neither speed nor slow the car. So purely a centripetal force keeps the speed constant.

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. . . . . the magnitude of the velocity stays constant is only true for the special case of circular motion with the body moving at constant speed with the force causing the acceleration at right angles to the velocity. In terms of energy the displacement of the body is at right angles to the force causing the acceleration, thus the force does no work and the kinetic energy of the body stays the same.

If the orbital motion is elliptical, parabolic or hyperbolic the force causing the acceleration is not at right angles to the velocity and the kinetic energy, and hence the speed of the body does not change.
In this case the force causing the acceleration does work and the kinetic energy, and hence speed, of the body changes.

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