so there are two forces acting on the P, a resorting force and an additional force that is forcing the system to oscillate.

The figure shows a particle P of mass 1kg which is free to slide horizontally inside a smooth cylindrical tube. The particle is attached to one end of a light elastic spring of natural length 0.5 m and modulus of elasticity 2 N.

The system is initially at rest. The other end Q of the spring is then forced to oscillate with simple harmonic motion so that at time t second its displacement from its initial position is meters.

The displacement of P from its initial positions at time seconds is meters, measured in the same direction as the displacement of Q

a) show that

b) Find the first time after the motion starts, at which P is instantaneously at rest.

the resorting force is due to the tension in the spring. by hookes law.

we are giving that the displace due to the addition force is so the acceleration due to this force at time is (by taking the derivative with respect to time twice)

both the resorting force and the additional force are opposing the direction of motion.

so by Newton's Second Law

which disagrees with the questions, who right me or them ???

For the next part I solved the differential equation.

I get the complementary function as

and the particular integral as

so I have

A is obviously zero as the particle had zero displacement initially and we are given that the derivative is zero at t = 0

so

so i must solve the equation so find out when this particle is at rest.

some simple trig gives

or

the smallest solution is given by . which agress with the book. So is my method good, or not ?

Many Thanks

Bobak