Question

The potential energy of a particle is determined by the expression $U=\alpha\left(x^{2}+y^{2}\right)$, where $\alpha$ is a positive constant. The particle begins to move from a point with the coordinates $(3,3)( m )$, only under the action of potential field force. Then its kinetic energy $T$ at the instant when the particle is at a point with the coordinates $(1,1)( m )$ is $32\, \alpha / n$. Find $n$.

Answer 1

$U =\alpha\left( x ^{2}+ y ^{2}\right)$
Total Mechanical energy $= KE + PE$
$=0+\alpha\left(3^{2}+3^{2}\right)$
$=18\, \alpha$
At $ x =1,1$
$PE =\alpha\left(1^{2}+1^{2}\right)=2 \alpha$
$KE + PE = C $
$T +2 \alpha =18 \alpha $
$T =16 \,\alpha$
$n =2$