2a. A mass moving back and forth on a spring obeys the sinusoidal equation x(t) = A cos(2πft), where A is the amplitude of the oscillation or the maximum displacement of the mass from its equilibrium position, f is the frequency of the oscillation and t is the time. If the amplitude is 10.0 cm and the frequency is f = 0.500 Hz, what is the position of the mass at t = 3.00 s? 2b. The velocity of the mass is also a sinusoidal function with the same period and frequency: v(t) = -vmax sin(2πft). Calculate vmax and the velocity of the mass v(t) when t = 3.00 s. 2c. What is the period T of this oscillation?
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.
2a. A mass moving back and forth on a spring obeys the sinusoidal equation x(t) = A cos(2πft), where A is the amplitude of the oscillation or the maximum displacement of the mass from its equilibrium position, f is the frequency of the oscillation and t is the time. If the amplitude is 10.0 cm and the frequency is f = 0.500 Hz, what is the position of the mass at t = 3.00 s?
2b. The velocity of the mass is also a sinusoidal function with the same period and frequency: v(t) = -vmax sin(2πft). Calculate vmax and the velocity of the mass v(t) when t = 3.00 s.
2c. What is the period T of this oscillation?
2d. If you evaluate the position function at t = 0 s the cosine of zero is 1 so you get the maximum amplitude of the oscillation, A. This occurs at the maximum stretch of the spring from its equilibrium position. For a mass on a spring this corresponds to the maximum spring potential energy, and because the velocity is zero at the endpoints of the oscillation, the kinetic energy at the endpoints is zero, so this value represents the total energy of the system: U(x) = 1⁄2 k x2 so U(x=A) = 1⁄2 k A2 If the spring constant k = 19.74 N/m, what is the total energy of this system?
2e. The maximum velocity occurs at the equilibrium position when the spring is not stretched. This position is x = 0, here the spring potential energy is zero and the kinetic energy is maximum and equal to the total energy of the system. What is the maximum speed of the mass if m = 2.00 kg? Hint: the kinetic energy is given by K = 1⁄2 mv2 andKmax =Umax =1⁄2kA2
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