A mass m = 1.4 kg hangs at the end of a vertical spring whose top end is fixed to the ceiling. The spring has spring constant k = 75 N/m and negligible mass. The mass undergoes simple harmonic motion when placed in vertical motion, with its position given as a function of time by y(t) = A cos(ωt – φ), with the positive y-axis pointing upward. At time t = 0 the mass is observed to be passing through its equilibrium height with an upward speed of v0 = 3.5 m/s. b. Find the smallest positive value of φ, in radians. φ=
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
A mass m = 1.4 kg hangs at the end of a vertical spring whose top end is fixed to the ceiling. The spring has spring constant k = 75 N/m and negligible mass. The mass undergoes
b. Find the smallest positive value of φ, in radians. φ=
c. Calculate the value of A, in meters. A=
d. What is the mass’s velocity along the y-axis, in meters per second, at time t1 = 0.25 s? v(t1) =
e. What is the magnitude of the mass’s maximum acceleration, in meters per second squared? amax=
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