The waveform shown is a signal x(t) = A cos(wt+ p). CAREFULLY determine the principal value of the phase, , expressed in RADIANS.
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- A mass of 55 grams stretches a spring by 8 cm. (Note that this means the forces balance, and thus mg = kx where m = 55 grams is mass, g = 981 cm/s is acceleration due to gravity, k is the spring constant, and x = 8 cm is the displacement.) The mass is set in motion from this equilibrium position with an initial downward velocity of 23 cm/s, and there is no damping. Find the position u (in cm) of the mass at any time t (in s). (Assume that position is measured upward from the equilibrium position.) u(t) Find the frequency (in radians per second), period (in seconds), and amplitude (in cm) of the motion. Frequency is Period is Amplitude isstraight line is always acted upon by a force from the centre of the path and directed Ex. 2: A particle of mass 8 gm moving in a and maximum acceleration, if the amplitude is towards the centre. Find its maximum velocity sse magnitude is 128 times the displacement 5 cm.Obtain the phasor notation of the following time-harmonic functions with an angular frequency of w (if it's possible): (a) V (t) = 6sin(wt – n/5), (b) V(t) = 20 cos(60nt – 60°). (c) I(t) = 2sin2 (wt) + 2cos2 (wt), (d) V (t) = sin(wt +n/3)sin(wt +1/6), (e) U(t) = -5 sin(wt)-2cos(@t), (f) D(t) = 1-sin(@t).
- Q.3 A counter-rotating eccentric mass exciter consisting of two rotating 400 g masses describing circles of 150 mm radius at the same speed, but in opposite senses, is placed on a machine element to induce a steady-state vibration. During a free vibration test, the assembly is displaced and released, the period of vibration is t s, and the ratio of consecutive amplitudes is r to 1.0. The spring stiffness is k = 550 N/m. When the speed of the exciter is 1200 rpm, determine: a) the amplitude of vibration b) the transmitted force to the ground exciter machine k 2 ||A magnesium atom (mass ≈≈ 24 proton masses) in a crystal is measured to oscillate with a frequency of roughly 1014 Hz. What is the effective spring constant of the forces holding the atom in the crystal?SA-1 The small angle approximation says that if 0 << 1 rad, then sin(0) ≈ 0, where is in radians. Recall that the % error in using this approximation is given by: % error approximate - exact exact x 100 (a) What is the % error in using the small angle approximation for the sine function, for an angle of 1 degree? (b) What is the % error in using the small angle approximation for the sine function, for an angle of 30 degrees? (c) What is the % error in using the small angle approximation for the sine function, for an angle of 80 degrees?
- Sine and cosine have period 2π.Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v) Prove tha tx(t0)=x(t0+2piw) In other words, the solution has the same value at time:t0 and at time:t0+2piw regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.An electron undergoes simple harmonic motion with the acceleration shown below: ax(t)=−amaxsin(2t/T) with amax=5839 ms2 and T=316 seconds. Assuming that the only motion is oscillatory (ignoring overall translation), what is the maximum speed of the electron? What is the amplitude of the electron's position?
- An undamped harmonic oscillator of mass m and spring constant k oscillates with an amplitude A. In terms of A, at what postion x is the speed of the oscillator, v, is half of its maximum speed v, max Hint: Set-up the energy equation at any postion. What is the maximum energy of the oscillator?Please help solve for v in (m/s)(a) An oscillating object repeats its motion every 3.3 seconds. (i) What is the period of this oscillation? (ii) What is its frequency? (iii) What is its phase rate (i.e. angular frequency)? (b) A magnesium atom (mass of 24 proton masses) in a crystal is measured to oscillate with a frequency of roughly 10l3 Hz. What is the effective spring constant of the forces holding that atom in the crystal? (c) Sitting on a trampoline, a person with mass m sinks a distance Az below the trampoline's normal level surface. (i) If the person gently bounces on the trampoline (without leaving its surface), what would be the person's period of oscillation T? (You should not need the person's mass, but if you think you do, assume and state a value.) (ii) Find T for Az = 45 cm. Check: For Az = 20 cm you should find T = 0.90 s.