A 3kg block is attached to an ideal spring with a
force constant k = 200N/m. The block is given
an initial velocity in the positive direction of
magnitude u = 12 m/s and no initial
displacement (x o = 0). Find (10 marks)
a. The amplitude and
b. The phase angle
c. Write an equation for the position as a
function of time
I will be happy to critique your work on this problem.
To find the amplitude of the motion, we can use the fact that the total mechanical energy (E) of the system is conserved. The mechanical energy is given by the sum of the potential energy (PE) stored in the spring and the kinetic energy (KE) of the block:
E = PE + KE
The potential energy of the spring is given by:
PE = (1/2)kx^2
where k is the force constant of the spring and x is the displacement from the equilibrium position.
The kinetic energy of the block is given by:
KE = (1/2)mv^2
where m is the mass of the block and v is the velocity of the block.
At the equilibrium position (x=0), all the initial energy is kinetic energy:
E = KE = (1/2)mv^2
Plugging in the values, we have:
E = (1/2)(3 kg)(12 m/s)^2 = 216 J
At the maximum displacement, all the initial energy is potential energy:
E = PE = (1/2)kA^2
where A is the amplitude of the oscillation.
Plugging in the values, we can solve for the amplitude:
216 J = (1/2)(200 N/m)A^2
A^2 = (216 J) / (100 N/m) = 2.16 m^2
A = sqrt(2.16) ≈ 1.47 m
Therefore, the amplitude of the oscillation is approximately 1.47 m.
To find the phase angle, we need to know the initial position (x_o) and velocity (v_o) of the block. In this case, the block is given no initial displacement (x_o = 0), so the phase angle (phi) is equal to zero.
Therefore, the phase angle of the oscillation is zero (phi = 0).
Finally, the equation for the position as a function of time can be expressed as:
x(t) = A cos(ωt + phi)
where A is the amplitude, ω is the angular frequency (ω = sqrt(k/m)), t is the time, and phi is the phase angle.
In this case, the equation becomes:
x(t) = (1.47 m) cos(sqrt(200 N/m / 3 kg) * t)
This equation describes the position of the block as a function of time.