A particle moves along the -axis under a piece-wise linear potential energy…

PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials

Problem

1.0\,\text{kg}

particle moves along the

x

-axis under a piece-wise linear potential energy distribution described below: • For

-3\,\text{m}\le x\le 0

V(x)=12+4x\;\text{J}

. • For

0\le x\le 3\,\text{m}

V(x)=12-2x\;\text{J}

. • For

|x|\ge 3\,\text{m}

V(x)

remains constant at its value at

x=\pm3\,\text{m}

. (a) Draw a qualitative graph of

V(x)

for

-5\,\text{m}\le x\le 5\,\text{m}

. Clearly label all key points. (b) Using the graph, write separate algebraic expressions for the force

F(x)

in each of the three regions defined above, and then sketch

F(x)

versus

x

over the same interval. (c) The particle has total mechanical energy

E=10\,\text{J}

. Mark on your

V(x)

graph the regions where motion is allowed and indicate all turning points. (d) From your sketch in part (b) discuss whether the equilibrium at

x=0

is stable, unstable, or neutral, giving one reason based on the behaviour of

F(x)

near this point.