A block of mass m is attached to a spring with spring constant k. The block is initially displaced a distance A from its equilibrium position and released from rest. The block undergoes simple harmonic motion.
(a) Express the angular frequency ω of the motion in terms of the mass m and the spring constant k.
(b) Determine the period T of the motion in terms of the angular frequency ω.
(c) How does the period T vary with the mass m when the spring constant k is kept constant? Justify your answer.
(d) How does the period T vary with the spring constant k when the mass m is kept constant? Justify your answer.
Provide detailed step-by-step explanations for each part.
(a) The angular frequency ω of the motion is given by:
ω = √(k / m)
The angular frequency of simple harmonic motion is derived from the relationship between the spring force F and the displacement x of the block. For a spring, F = -kx, and using Newton's second law, F = ma, where a is the acceleration of the block.
Setting those two equations equal, we have -kx = ma. Since a = d²x / dt², we can rewrite the equation as:
-kx = md²x / dt²
Dividing both sides by -m and rearranging, we find:
d²x / dt² = - k / m * x
This equation is the equation of simple harmonic motion, and its solution is x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase constant.
By comparing this equation with the equation derived from Newton's second law, we find that ω = √(k / m).
(b) The period T of the motion is the time taken for one complete oscillation. It is related to the angular frequency ω by the equation:
T = 2π / ω
This equation follows from the fact that one complete oscillation is equivalent to a 2π change in the argument of the cosine function.
(c) When the spring constant k is kept constant, the angular frequency ω remains constant. From the equation T = 2π / ω, we can see that the period T is inversely proportional to the angular frequency ω. Therefore, when ω is constant, the period T will also be constant. The period does not depend on the mass m when k is kept constant.
(d) When the mass m is kept constant, the angular frequency ω varies inversely with the square root of the spring constant k. From the equation T = 2π / ω, we can see that the period T is directly proportional to the angular frequency ω. Therefore, when ω varies inversely with the square root of k, the period T will also vary inversely with the square root of k. As the spring constant k increases, the period T decreases, and vice versa, when m is kept constant.
[Source: College Board AP Physics 1 Course and Exam Description]