3) A 5 kg mass is attached to a spring with spring constant k = 100 N/m, and moves along a horizontal surface which produces a resistive force R = -bv where b = 8 N/m/s. a. Calculate what the angular frequency co would be if there were no resistive force/damping. Is the damped angular frequency greater of less than this? b. Calculate the damping constant a, and what type of damped motion (under critical or over critical damped) occurs? c. Calculate the actuals (damped) angular frequency of the motion. d. The position of the object at any time is x(t) = e-at [Pcos(wot) + Qsin(wot)], where P and Q are constants. Differentiate this to obtain its velocity at any time. e. If the object is released from rest at 1=0 at position xo=0.1 m, find the values of P and Q. f. Convert your solution to the form x(t) = Ae-at cos(wot-d) by finding the values of A and from P and Q. How long does it take the mass to reach the equilibrium point (x = 0) for the first time? g. Differentiate your solution in the new form to get the velocity. How long does is take the mass to reach its furthest distance from the starting point, and how far is this maximum distance from the initial position? [Hint: v=0 at the furthest point] h. Calculate the energy of the system, firstly at its initial point, secondly as it passes through the equilibrium position for the first time (you will have to calculate the velocity there), and thirdly as it reaches the point furthest from its initial point. i. Using the expression for the average energy of an under critically damped harmonic oscillator, calculate how long it takes before the average energy of the system is halved j. Calculate the relaxation time, the logarithmic decrement and the quality factor of the system.
3) A 5 kg mass is attached to a spring with spring constant k = 100 N/m, and moves along a horizontal surface which produces a resistive force R = -bv where b = 8 N/m/s. a. Calculate what the angular frequency co would be if there were no resistive force/damping. Is the damped angular frequency greater of less than this? b. Calculate the damping constant a, and what type of damped motion (under critical or over critical damped) occurs? c. Calculate the actuals (damped) angular frequency of the motion. d. The position of the object at any time is x(t) = e-at [Pcos(wot) + Qsin(wot)], where P and Q are constants. Differentiate this to obtain its velocity at any time. e. If the object is released from rest at 1=0 at position xo=0.1 m, find the values of P and Q. f. Convert your solution to the form x(t) = Ae-at cos(wot-d) by finding the values of A and from P and Q. How long does it take the mass to reach the equilibrium point (x = 0) for the first time? g. Differentiate your solution in the new form to get the velocity. How long does is take the mass to reach its furthest distance from the starting point, and how far is this maximum distance from the initial position? [Hint: v=0 at the furthest point] h. Calculate the energy of the system, firstly at its initial point, secondly as it passes through the equilibrium position for the first time (you will have to calculate the velocity there), and thirdly as it reaches the point furthest from its initial point. i. Using the expression for the average energy of an under critically damped harmonic oscillator, calculate how long it takes before the average energy of the system is halved j. Calculate the relaxation time, the logarithmic decrement and the quality factor of the system.
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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.
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![3) A 5 kg mass is attached to a spring with spring constant k = 100 N/m, and moves along a horizontal
surface which produces a resistive force R = -bv where b = 8 N/m/s.
a. Calculate what the angular frequency o would be if there were no resistive force/damping.
Is the damped angular frequency greater of less than this?
b. Calculate the damping constant a, and what type of damped motion (under critical or over
critical damped) occurs?
c. Calculate the actuals (damped) angular frequency of the motion.
d. The position of the object at any time is x(t) = e-at [Pcos (wot) + Qsin(wot)], where P
and Q are constants. Differentiate this to obtain its velocity at any time.
f.
e. If the object is released from rest at 1 = 0 at position xo = 0.1 m, find the values of P and Q.
Convert your solution to the form x(t) = Ae-at cos(wot-d) by finding the values of A
and from P and Q. How long does it take the mass to reach the equilibrium point (x = 0)
for the first time?
g. Differentiate your solution in the new form to get the velocity. How long does is take the
mass to reach its furthest distance from the starting point, and how far is this maximum
distance from the initial position? [Hint: v=0 at the furthest point]
h. Calculate the energy of the system, firstly at its initial point, secondly as it passes through
the equilibrium position for the first time (you will have to calculate the velocity there),
and thirdly as it reaches the point furthest from its initial point.
i. Using the expression for the average energy of an under critically damped harmonic
oscillator, calculate how long it takes before the average energy of the system is halved
j. Calculate the relaxation time, the logarithmic decrement and the quality factor of the
system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85d28585-13ce-49e4-ac80-6225c01be022%2F85e0d781-dcf3-4378-b2ab-7469c489f094%2Fvx1ldne_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3) A 5 kg mass is attached to a spring with spring constant k = 100 N/m, and moves along a horizontal
surface which produces a resistive force R = -bv where b = 8 N/m/s.
a. Calculate what the angular frequency o would be if there were no resistive force/damping.
Is the damped angular frequency greater of less than this?
b. Calculate the damping constant a, and what type of damped motion (under critical or over
critical damped) occurs?
c. Calculate the actuals (damped) angular frequency of the motion.
d. The position of the object at any time is x(t) = e-at [Pcos (wot) + Qsin(wot)], where P
and Q are constants. Differentiate this to obtain its velocity at any time.
f.
e. If the object is released from rest at 1 = 0 at position xo = 0.1 m, find the values of P and Q.
Convert your solution to the form x(t) = Ae-at cos(wot-d) by finding the values of A
and from P and Q. How long does it take the mass to reach the equilibrium point (x = 0)
for the first time?
g. Differentiate your solution in the new form to get the velocity. How long does is take the
mass to reach its furthest distance from the starting point, and how far is this maximum
distance from the initial position? [Hint: v=0 at the furthest point]
h. Calculate the energy of the system, firstly at its initial point, secondly as it passes through
the equilibrium position for the first time (you will have to calculate the velocity there),
and thirdly as it reaches the point furthest from its initial point.
i. Using the expression for the average energy of an under critically damped harmonic
oscillator, calculate how long it takes before the average energy of the system is halved
j. Calculate the relaxation time, the logarithmic decrement and the quality factor of the
system.
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