A 1.2 kg block is attached to a horizontal 23 N/m spring and resting on a frictionless

horizontal surface. The block is set oscillating with amplitude 10 cm and phase constant –
π/2. A 2nd block with mass 0.8 kg moving at 1.7 m/s collides inelastically (the blocks stick
together) with the 1.2 kg block when it is at its maximum displacement. Determine the
frequency, amplitude, and phase constant (relative to the original t = 0) of the resulting
motion.

m2•v2 =(m1+m2) •v

v= m2•v2/(m1+m2) = 0.8•1.7/(1.2+0.8) =0.68 m/s = v(max)
ω =sqrt{k/(m1+m2)} = sqrt{23/2) = 3.4 rad/s.
v(max) = A1• ω,
A1 = v(max)/ ω = 0.68/3.4 =0.2 m.
ω = 2•π•f,
f = ω/ 2•π = 3.4/2•3.14 =0.54 Hz.

To determine the frequency, amplitude, and phase constant of the resulting motion after the collision between the two blocks, we can follow these steps:

Step 1: Find the natural frequency (ω) of the original block before the collision.
The formula for the natural frequency of a mass-spring system is ω = √(k/m), where k is the spring constant and m is the mass attached to the spring.

Given: k = 23 N/m, m = 1.2 kg
Using the given values, we can calculate the natural frequency:
ω = √(23 N/m / 1.2 kg) = √19.17 rad/s ≈ 4.38 rad/s

Step 2: Find the new mass and velocity of the combined blocks after the collision.
The total mass after the collision is the sum of the masses of the two blocks: m_total = 1.2 kg + 0.8 kg = 2 kg.
To find the new velocity (v_new) of the combined blocks after the collision, we can use the principle of conservation of momentum:
m1 * v1_initial + m2 * v2_initial = (m1 + m2) * v_new,
where m1 and m2 are the masses of the blocks, and v1_initial and v2_initial are their initial velocities.

Given: m1 = 1.2 kg, v1_initial = 0 (as it is at maximum displacement), m2 = 0.8 kg, and v2_initial = 1.7 m/s,
Using the given values, we can calculate the new velocity after the collision:
(1.2 kg * 0) + (0.8 kg * 1.7 m/s) = (2 kg) * v_new,
0 + 1.36 kg⋅m/s = 2 kg * v_new,
v_new = 1.36 kg⋅m/s / 2 kg = 0.68 m/s.

Step 3: Find the amplitude (A) and phase constant (φ) of the resulting motion.
To find the amplitude and phase constant of the resulting motion, we need to express the initial displacement of the system as a function of time. The equation of motion for a mass-spring system with simple harmonic motion is:
x(t) = A * cos(ω * t + φ),
where x(t) is the displacement at time t, A is the amplitude, ω is the angular frequency (ω = 2πf, where f is the frequency), t is the time, and φ is the phase constant.

Given: Amplitude of the original motion = 10 cm = 0.1 m, Phase constant of the original motion = -π/2.
We can now write the equation for the resulting motion after the collision:
x(t) = A * cos(ω * t + φ).

Substituting the known values, we get:
x(t) = A * cos(4.38 rad/s * t - π/2),

Comparing this equation to the equation of motion, we can conclude:

Amplitude (A) of the resulting motion = Amplitude of the original motion = 0.1 m.
Phase constant (φ) of the resulting motion = Phase constant of the original motion = -π/2.

To find the frequency of the resulting motion, we can use the relationship between angular frequency (ω) and frequency (f) as follows:
ω = 2πf,
4.38 rad/s = 2πf,
f = 4.38 rad/s / 2π ≈ 0.696 Hz.

Therefore, the frequency of the resulting motion is approximately 0.696 Hz.