Problem — True or false on work and energy

Problem

State whether the following statements are true or false, justifying your answer: (a) the work done by a force perpendicular to the displacement is always zero; (b) kinetic energy can be negative; (c) if the resultant of the forces is zero, the total work is zero; (d) two objects with the same momentum always have the same kinetic energy.

Solution

The work done by a constant force is $L = F\,s\,\cos\theta$, where $\theta$ is the angle between the force and the displacement; kinetic energy is $E_k = \tfrac{1}{2}mv^2$.

(a) TRUE. If the force is perpendicular to the displacement, $\theta = 90^\circ$ and $\cos 90^\circ = 0$, so $L = 0$. This is the case, for example, of the centripetal force in uniform circular motion or of the normal force on a horizontal plane: they do no work.

(b) FALSE. Kinetic energy is $E_k = \tfrac{1}{2}mv^2$: positive mass and a non-negative square of the speed always give $E_k \geq 0$. It cannot be negative.

(c) TRUE. If the resultant $\vv{F}$ is zero, the total work is that of the resultant, $L_\text{tot} = \vv{F}\cdot\vv{s} = 0$. Consistently, by the work-energy theorem $L_\text{tot} = \Delta E_k = 0$: with no net force the speed (in magnitude) does not change.

(d) FALSE. With momentum $p = mv$ we have $E_k = \tfrac{1}{2}mv^2 = \dfrac{p^2}{2m}$. For equal $p$, the kinetic energy depends on the mass: the lighter object has greater kinetic energy. Equal $p$ does not imply equal $E_k$.

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

Links

Topics: Work and energy Concepts: Work done by a force · Kinetic energy Skills: Ranking reasoning