A mass m1 = 8.51 kg𝑚1 = 8.51 kg is at rest on a frictionless horizontal surface and connected to a wall by a spring with k = 70.1 N/m, as shown in the figure. A second mass m2 = 5.71 kg𝑚2 = 5.71 kg is moving to the right at v0 = 15.1 m/s𝑣0 = 15.1 m/s . The two masses collide and stick together. How long will it take after the collision to reach this maximum compression?
Question
A mass m1 = 8.51 kg𝑚1 = 8.51 kg is at rest on a frictionless horizontal surface and connected to a wall by a spring with k = 70.1 N/m, as shown in the figure. A second mass m2 = 5.71 kg𝑚2 = 5.71 kg is moving to the right at v0 = 15.1 m/s𝑣0 = 15.1 m/s . The two masses collide and stick together. How long will it take after the collision to reach this maximum compression?
Solution
The problem involves a collision followed by a simple harmonic motion.
First, we need to find the velocity of the combined masses immediately after the collision. This can be found using the principle of conservation of momentum.
The initial momentum of the system is given by the momentum of the moving mass m2, which is m2*v0. The final momentum of the system is the combined mass (m1 + m2) times the final velocity v.
Setting these equal gives:
m2*v0 = (m1 + m2)*v
Solving for v gives:
v = m2*v0 / (m1 + m2)
Next, we need to find the maximum compression of the spring. This occurs when the kinetic energy of the masses is completely converted into potential energy in the spring.
The initial kinetic energy is (1/2)*(m1 + m2)*v^2 and the potential energy in the spring at maximum compression is (1/2)kx^2, where x is the maximum compression.
Setting these equal gives:
(1/2)*(m1 + m2)*v^2 = (1/2)kx^2
Solving for x gives:
x = sqrt((m1 + m2)*v^2 / k)
Finally, we need to find the time it takes to reach this maximum compression. The motion of the masses is a simple harmonic motion, so the time can be found using the period of the motion.
The period T of a mass-spring system is given by T = 2pisqrt(m/k), where m is the total mass (m1 + m2) and k is the spring constant.
The time it takes to reach maximum compression is a quarter of the period, so:
t = T/4 = (1/4)2pi*sqrt((m1 + m2) / k)
Substitute the given values into the above equations to find the numerical answers.
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