At equilibrium, a mass M = 0.3 kg stretches a spring by L = 0.1 m. The mass is initially released from a point below the equilibrium position by a distance x0 = 0.1 m with a downwards velocity v0 = 1.5 ms-1. For the gravity of acceleration, assume g = 10 ms-2 (a) Calculate the equation of motion x(t) as function of t only: use the numerical values given above for all other parameters in SI units. Write the resulting equation using Maple syntax, e.g. 1/10*cos(3*t)+2/5*sin(4*t). Do not approximate numbers. For instance write 1/3 instead of 0.333 and sqrt(2) instaed of 1.414. (b) Write the amplitude A and the phase angle ϕϕ (in radians) of the motion. Write your answers as decimals correct to the third decimal place (c) How many completed cycles N will the mass have completed at the end of 3π seconds? (d) At what time t does the mass pass through the equilibrium point for the second time going upwards? (Hint: do not forget that the x-axis points downwards, see Figure). Write your answers as decimals correct to the third decimal. (e) What is the position of the mass at t = 3 s?
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.
At equilibrium, a mass M = 0.3 kg stretches a spring by L = 0.1 m.
The mass is initially released from a point below the equilibrium position by a distance x0 = 0.1 m with a downwards velocity v0 = 1.5 ms-1. For the gravity of acceleration, assume g = 10 ms-2
(a) Calculate the equation of motion x(t) as function of t only: use the numerical values given above for all other parameters in SI units. Write the resulting equation using Maple syntax, e.g. 1/10*cos(3*t)+2/5*sin(4*t). Do not approximate numbers. For instance write 1/3 instead of 0.333 and sqrt(2) instaed of 1.414.
(b) Write the amplitude A and the phase angle ϕϕ (in radians) of the motion. Write your answers as decimals correct to the third decimal place
(c) How many completed cycles N will the mass have completed at the end of 3π seconds?
(d) At what time t does the mass pass through the equilibrium point for the second time going upwards? (Hint: do not forget that the x-axis points downwards, see Figure). Write your answers as decimals correct to the third decimal.
(e) What is the position of the mass at t = 3 s?
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