An object moving along a horizontal track collides with and compresses a light spring (which obeys Hooke's Law) located at the end of the track. The spring constant is 38.6 N/m, the mass of the object 0.270 kg and the speed of the object is 1.20 m/s immediately before the collision. (a) Determine the spring's maximum compression if the track is frictionless. (b) If the track is not frictionless, will the spring's maximum compression be greater than, less than, or equal to the value obtained in part (a)?
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To solve this problem, we can use the principle of conservation of mechanical energy. Initially, the object has only kinetic energy, and after the collision, the energy is stored as potential energy in the compressed spring.
(a) Let's find the maximum compression of the spring assuming a frictionless track.
1. Calculate the initial kinetic energy of the object:
Kinetic energy = (1/2) * mass * velocity^2
= (1/2) * 0.270 kg * (1.20 m/s)^2
2. Calculate the potential energy stored in the spring when it reaches its maximum compression:
Potential energy = (1/2) * spring constant * maximum compression^2
Equating the initial kinetic energy to the potential energy, we have:
(1/2) * 0.270 kg * (1.20 m/s)^2 = (1/2) * 38.6 N/m * maximum compression^2
Solving for maximum compression:
maximum compression^2 = (0.270 kg * (1.20 m/s)^2) / (38.6 N/m)
maximum compression = sqrt((0.270 kg * (1.20 m/s)^2) / (38.6 N/m))
Calculate the maximum compression.
(b) If the track is not frictionless, some of the initial kinetic energy will be lost due to friction. Therefore, the spring's maximum compression will be less than the value obtained in part (a) because some mechanical energy is lost to friction.
So, the spring's maximum compression in part (b) will be less than the value obtained in part (a).
To determine the spring's maximum compression, we can use the principles of conservation of energy.
(a) If the track is frictionless, the initial kinetic energy of the object will be transferred to the elastic potential energy stored in the spring when it is compressed.
The initial kinetic energy of the object, Ek, is given by:
Ek = (1/2)mv^2
where:
m = mass of the object = 0.270 kg
v = velocity of the object = 1.20 m/s
Ek = (1/2)(0.270 kg)(1.20 m/s)^2
Ek = 0.1944 J
The elastic potential energy stored in the compressed spring, Ep, can be calculated using Hooke's Law:
Ep = (1/2)kx^2
where:
k = spring constant = 38.6 N/m
x = compression of the spring (maximum compression)
Setting Ek equal to Ep:
0.1944 J = (1/2)(38.6 N/m)x^2
Solving for x^2:
0.3888 J/Nm = x^2
x = √(0.3888 J/Nm) or x = ±0.6233 m (maximum compression is a positive value)
Therefore, the spring's maximum compression is approximately 0.6233 m.
(b) If the track is not frictionless, some energy will be lost to friction during the collision. This means that the total energy transferred to the spring will be less than in the frictionless case. Consequently, the spring's maximum compression will be less than the value obtained in part (a).
a. compresses spring energy=initial KE
b. max compression will be less, of course.