First slide

Conservation of mechanical energy

Question

A single conservative force acts on a 1 kg particle that moves along x-axis. The potential energy of the particle varies with x as U = 20+ (x - 2)² here U is in joules and x is in meters. When the particle is at x = 5 m, its kinetic energy is 20 J. Then which of the following is/are correct ?

Moderate

A

Mechanical energy of particle is 49 J.
B

Least and greatest value of x between which particle can move is (2 - $\sqrt{29}$ ) m and (2+ $\sqrt{29}$ ) m respectively.
C

Maximum kinetic energy of the particle is 29 J.
D

At x = 2, the body is in equilibrium.

Solution

$\begin{array}{l} U = 20 + (x - 2)^{2} \\ (a) At x = 5, U = 20 + 3^{2} = 29 J \\ K E = 20 J \end{array}$
$∴ Mechanical energy = U + KE = 49 J$
$\begin{array}{l} (c) U_{min} = 20 J \\ ∴ {KE}_{max} = 29 J \\ (b) {KE}_{min} = 0 \\ ∴ U_{max} = 49 = 20 + {(x_{max} - 2)}^{2} \\ \Rightarrow {(x_{max} - 2)}^{2} = 29 \\ \Rightarrow x_{max} = 2 \pm \sqrt{29} \end{array}$
$(d) For equilibrium \frac{dU}{dx} = 0$
$\Rightarrow x = 2$

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