Final answer:
the motion of a mass on a nonideal spring with a restoring force of -kx-ax^3 is not simple harmonic in all cases. The period of the motion depends on the specific form of the restoring force and can be found using the modified frequency that includes both k and a. In case B, where the cubic term is ignored, the motion is simple harmonic. In cases A, C, and D, where the cubic term is present or a constant force is added to the restoring force, the motion is not simple harmonic.The correct option is c.
Step-by-step explanation:
Let's analyze each case:
A) The restoring force provided by the nonideal spring is -kx-ax^3. The linear term (-kx) represents the behavior of a typical ideal spring, while the cubic term (-ax^3) adds a nonlinear component. Since the cubic term is not proportional to x, this is not a simple harmonic motion. The period of the motion can be found by setting the acceleration equal to zero and finding the frequency, then using the formula for period (T=2π/ω). This results in:
ω = √(k/m) + 3a/2m
T = 2π/ω
B) If we ignore the cubic term (-ax^3), we're left with a restoring force of -kx, which is proportional to x and represents simple harmonic motion. The period and frequency can be found using the formula for simple harmonic motion (T=2π/ω, ω=√(k/m)).
C) In this case, we've added a constant force (-3ax) to the restoring force (-kx). This means that the motion is no longer simple harmonic, as the acceleration is not proportional to position. The period can still be found using the formula for T=2π/ω, but with a modified frequency that includes both k and -3a.
D) Similar to case C, we've added a constant force (-3ax) to the restoring force (-kx), but this time we've divided it by 2m instead of adding it directly. This results in a different modified frequency and period compared to case C. Again, since the acceleration is not proportional to position, this is not simple harmonic motion.The correct option is c.