True or false on circular motion and inclines

Problem

True or false, giving reasons: (a) in uniform circular motion the work done by the centripetal force is zero; (b) on a frictionless incline, the acceleration of a body does not depend on its mass; (c) centrifugal force is a fundamental force of nature; (d) at the top of the “loop-the-loop” the minimum speed depends on the mass of the body.

Solution

(a) TRUE. The centripetal force is always directed towards the centre, hence perpendicular to the velocity (which is tangent to the trajectory). The work is $L = \vv{F}\cdot\vv{s}$ and, since force and displacement are perpendicular at every instant, $L = 0$. This is consistent with the fact that in uniform circular motion the magnitude of $v$ (and hence the kinetic energy) does not change.

(b) TRUE. On the frictionless incline, along the incline we have $ma = mg\sin\theta$, from which $a = g\sin\theta$ The mass cancels: the acceleration depends only on the angle (and on $g$), not on $m$. This is the analogue, on the incline, of the mass-independence of free fall.

(c) FALSE. Centrifugal force is not a fundamental force. It is an apparent (or inertial, or fictitious) force that appears only when the motion is described from a rotating reference frame, i.e. a non-inertial one. In the inertial frame it does not exist: there is only the real centripetal force (tension, friction, gravity…) that bends the trajectory. The fundamental forces are gravitational, electromagnetic, strong nuclear and weak nuclear.

(d) FALSE. At the top of the loop-the-loop, the limiting condition is that the weight force alone provides the centripetal force ($N \to 0$): $mg = \frac{m v_\text{min}^2}{r} \;\Rightarrow\; v_\text{min} = \sqrt{g\,r}$ The mass cancels: the minimum speed depends only on $g$ and the radius $r$, not on the mass of the body.

$\ev{\text{(a) T}\quad\text{(b) T}\quad\text{(c) F}\quad\text{(d) F}}$

Links

Topics: Dynamics Concepts: Centripetal force · Uniform circular motion · Newton’s second law · Weight force Skills: Applying Newton’s laws · Analysis of limiting cases and “fantaphysics” Objects: Inclined plane · Loop-the-loop