A block having the mass m and the charge +Q is connected to an insulating spring

Question

A block having the mass m and the charge +Q is connected to an insulating spring having a force constant k. The block lies on a frictionless, insulating, horizontal track, and the system is immersed in a uniform electric filed of magnitude E directed away from the spring. The block is released from rest when the spring is unstretched (at x=0). We wish to show that the ensuing motion of the block is simple harmonic. Call the initial configuration of the system that existing just as the block is released from rest. The final configuration is when the block momentarily comes to rest again. What is the value of x when the block comes to rest momentarily? please show all work and calculations

Explanation / Answer

For part C, use the conservation of energy to set up an equation:

Ki + Ui = Kf +Uf

and substitute in corresponding components:

0 + QVi = 0 + (1/2 * k*x^2 + QVf)

so simplify to get

1/2 * k*x^2 +Q*E*x

Now solve for x:

x = (2*Q*E) / k

For part D the answer is particle in equilibrium because there is zero net force on it.

Part E, use conclusion from part D, to solve:

The two forces are the spring and the electric force, and we know that

Fs = -kx and Fe = EQ

Fs + Fe = 0

so Fs = - Fe

substitute in values and solve for x:

-kx = -EQ

and so x = EQ / k

Part G, the equation for tension is T = 2pi / angular frequency

and angular frequency = squareroot of (k/m)

so T = 2pi / squareroot of (k/m)

simplifying you get that

T = 2pi * squareroot of (m/k)

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