In the lecture note, the resonance of a forced oscillation is defined as a state where...
In the lecture note, the resonance of a forced oscillation is defined as a state where the amplitude of displacement reaches maximum. Resonance can also be defined for the acceleration, where the acceleration’s amplitude reaches maximum. (a) Find the frequency of acceleration resonance. (b) Assume the damping is very small, i.e. ? ≪ ?0, find the approximate expression of the frequency in (a) [Hint: using Taylor series] and compare it with the frequency of displacement resonance.
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In the lecture note, the resonance of a forced oscillation is
defined as a state where...
(1 point) This is an example of an Undamped Forced Oscillation where the phenomenon of Beats...
(1 point) This is an example of an Undamped Forced Oscillation where the phenomenon of Beats Occurs. Find the solution of the initial value problem: " +7.84z-5cos(3), (0) '(0) 0 Graph the solution to confirm the phenomenon of Beats. Note that you may have to use a large window in order to see more than one beat. What is the length of each beat? Length Would you be able to explain why the beats phenomenon occurs for this particular example?...
Design dala Observalion deck mass m-25,000 k Danong ratio 0.5% Figure 91. Determine the equation of motion ofthe ๒wer teevibraorntheform (15 marks) mitt) + car)+xt)- where xt) is the horizontal displ...
Design dala Observalion deck mass m-25,000 k Danong ratio 0.5% Figure 91. Determine the equation of motion ofthe ๒wer teevibraorntheform (15 marks) mitt) + car)+xt)- where xt) is the horizontal displacement of the top of the tower b) Determine the damped natural frequency, fa (in Hz) of the tower (10 marks) ) A radar device, which inckdes a large rotaling eccentic mass, has been (30 marks) nstalled at the top of the tower Unfortunately, it has a trequency of rotation...
PROBLEM 5. TUNING A CIRCUIT: PRACTICAL RESONANCE. Consider a forced RLC circuit with L-1 (H), R-1...
PROBLEM 5. TUNING A CIRCUIT: PRACTICAL RESONANCE. Consider a forced RLC circuit with L-1 (H), R-10 (12) and C 丽0 (f). Suppose an alternating current supplies a electromotive force Et)100 coswt. The equation modeling the charge Q(t) on the capacitor is 650 Q"(t) 10Q650Q(t) 100 coswt. a. Is the damping over-, under- or critical? Find the form of the general solution. Identify the transient and steady-state parts of the solution. b. Find the amplitude C(w) of the steady-state piece (here...
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest...
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest at X +0.750 m. The oscillator is set into motion once again from x=+0.750 m, except the oscillator now experiences a retarding force that is linear with respect to velocity. As a result, the oscillator does not return to its original starting position, but instead reaches = +0.700 m after one period. a. During the first full oscillation of motion, determine the fraction of...
Learning Goal: To understand the use of phasor diagrams in calculating the impedance and resonance conditions in a seri...
Learning Goal: To understand the use of phasor diagrams in calculating the impedance and resonance conditions in a series L-R-C circuit. At resonance, XL = Xc. The voltage across the capacitor exactly cancels that across the i have the same amplitude. Thus, the inductor and capacitor effectively cancel out in the formu not come as a surprise that the resonant frequency equals the natural frequency of the oscilla In this problem, you will consider a series L-R-C circuit, containing a...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
A pendulum consists of a string of length L and a mass m hung at one...
A pendulum consists of a string of length L and a mass m hung at one end and the mass oscillates along a circular arc. Part a) Familiarize yourself with the derivation of omega = Squareroot g/L to hold. i) Explain succinctly how the angular frequency of oscillation omega = Squareroot g/L comes about from Newton's Law, where g is the gravitational acceleration. ii) One assumption required is the small angle approximation: sin theta = theta and cos theta =...
please answer as many questions as possible. I will “thumb up” the answers. Thanks! 1. You are on a boat, which is bobbing up and down. The boat's vertical displacement y is given by y 1.2...
please answer as many questions as possible. I will “thumb up” the
answers. Thanks!
1. You are on a boat, which is bobbing up and down. The boat's vertical displacement y is given by y 1.2 cos(t). Find the amplitude, angular frequency, phase constant, frequency, and period of the motion. (b) Where is the boat at t 1 s? (c) Find the velocity and acceleration as functions of time t. (d) Find the initial values of the position, velocity, and...
Experimental: Freq (Hz) VL+RL (V) VR(V) VC(V) 600 0.167 0.249 1.09 700 0.228 0....
Experimental:
Freq (Hz)
VL+RL (V)
VR(V)
VC(V)
600
0.167
0.249
1.09
700
0.228
0.229
1.123
800
0.302
0.352
1.158
900
0.398
0.417
1.217
1000
0.504
0.481
1.263
1100
0.626
0.547
1.305
1200
0.763
0.613
1.342
1300
0.905
0.674
1.361
1400
1.042
0.723
1.356
1500
1.163
0.754
1.322
1600
1.261
0.768
1.246
1700
1.326
0.761
1.177
1800
1.363
0.741
1.081
1900
1.375
0.71
0.982
2000
1.375
0.675
0.887
2100
1.364
0.639
0.799
2200
1.348
0.603
0.721
2300
1.33
0.569
0.65
2400...
2. Dipole-dipole force The charge distribution where two equal but opposite charges are separated by a...
2. Dipole-dipole force The charge distribution where two equal but opposite charges are separated by a fixed distance a dipole and is very common and important in nature. Consider two dipoles, each consisting of charges ±q separated by a distance δ. The axes of two dipoles are parallel and their midpoints are separated by a distance r. One is inverted compared to the other. See the figure tq +9 (a) Calculate the force exerted on each dipole by the other...
(1 point) This is an example of an Undamped Forced Oscillation where the phenomenon of Beats Occurs. Find the solution of the initial value problem: " +7.84z-5cos(3), (0) '(0) 0 Graph the solution to confirm the phenomenon of Beats. Note that you may have to use a large window in order to see more than one beat. What is the length of each beat? Length Would you be able to explain why the beats phenomenon occurs for this particular example?...
Design dala Observalion deck mass m-25,000 k Danong ratio 0.5% Figure 91. Determine the equation of motion ofthe ๒wer teevibraorntheform (15 marks) mitt) + car)+xt)- where xt) is the horizontal displacement of the top of the tower b) Determine the damped natural frequency, fa (in Hz) of the tower (10 marks) ) A radar device, which inckdes a large rotaling eccentic mass, has been (30 marks) nstalled at the top of the tower Unfortunately, it has a trequency of rotation...
PROBLEM 5. TUNING A CIRCUIT: PRACTICAL RESONANCE. Consider a forced RLC circuit with L-1 (H), R-10 (12) and C 丽0 (f). Suppose an alternating current supplies a electromotive force Et)100 coswt. The equation modeling the charge Q(t) on the capacitor is 650 Q"(t) 10Q650Q(t) 100 coswt. a. Is the damping over-, under- or critical? Find the form of the general solution. Identify the transient and steady-state parts of the solution. b. Find the amplitude C(w) of the steady-state piece (here...
1. An ideal (frictionless) simple harmonic oscillator is set into motion by releasing it from rest at X +0.750 m. The oscillator is set into motion once again from x=+0.750 m, except the oscillator now experiences a retarding force that is linear with respect to velocity. As a result, the oscillator does not return to its original starting position, but instead reaches = +0.700 m after one period. a. During the first full oscillation of motion, determine the fraction of...
Learning Goal: To understand the use of phasor diagrams in calculating the impedance and resonance conditions in a series L-R-C circuit. At resonance, XL = Xc. The voltage across the capacitor exactly cancels that across the i have the same amplitude. Thus, the inductor and capacitor effectively cancel out in the formu not come as a surprise that the resonant frequency equals the natural frequency of the oscilla In this problem, you will consider a series L-R-C circuit, containing a...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
A pendulum consists of a string of length L and a mass m hung at one end and the mass oscillates along a circular arc. Part a) Familiarize yourself with the derivation of omega = Squareroot g/L to hold. i) Explain succinctly how the angular frequency of oscillation omega = Squareroot g/L comes about from Newton's Law, where g is the gravitational acceleration. ii) One assumption required is the small angle approximation: sin theta = theta and cos theta =...
please answer as many questions as possible. I will “thumb up” the
answers. Thanks!
1. You are on a boat, which is bobbing up and down. The boat's vertical displacement y is given by y 1.2 cos(t). Find the amplitude, angular frequency, phase constant, frequency, and period of the motion. (b) Where is the boat at t 1 s? (c) Find the velocity and acceleration as functions of time t. (d) Find the initial values of the position, velocity, and...
Experimental:
Freq (Hz)
VL+RL (V)
VR(V)
VC(V)
600
0.167
0.249
1.09
700
0.228
0.229
1.123
800
0.302
0.352
1.158
900
0.398
0.417
1.217
1000
0.504
0.481
1.263
1100
0.626
0.547
1.305
1200
0.763
0.613
1.342
1300
0.905
0.674
1.361
1400
1.042
0.723
1.356
1500
1.163
0.754
1.322
1600
1.261
0.768
1.246
1700
1.326
0.761
1.177
1800
1.363
0.741
1.081
1900
1.375
0.71
0.982
2000
1.375
0.675
0.887
2100
1.364
0.639
0.799
2200
1.348
0.603
0.721
2300
1.33
0.569
0.65
2400...
2. Dipole-dipole force The charge distribution where two equal but opposite charges are separated by a fixed distance a dipole and is very common and important in nature. Consider two dipoles, each consisting of charges ±q separated by a distance δ. The axes of two dipoles are parallel and their midpoints are separated by a distance r. One is inverted compared to the other. See the figure tq +9 (a) Calculate the force exerted on each dipole by the other...
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