Physics: Oscillations
Chapter 14: Oscillations Chapter 15: Wave Motion From 'Physics for scientists and engineers' by Giancoli.
40 flashcards · Physics · by lakesare
A mass is oscillating on a frictionless surface at the end of a horizontal spring. Where, if anywhere, is acceleration of the mass zero?
At x = 0, - where F is zero.
If an oscillating mass has a frequency of 1.25 Hz, it makes 100 oscillations in how many seconds?
V = 1.25 oscillations/s S = 100 oscillations t = 100/1.25 = 80s
When a family of four with a total mass of 200 kg step into their 1200-kg car, the car’s springs compress 3.0 cm. What is the spring constant of the car's springs?
The oscillation of a spring hung vertically is essentially the same as that of a horizontal spring. Because of gravity, the length of a vertical spring with a mass m on the end will be longer at equilibrium than when that same spring is horizontal, as shown in Fig. 14-3. The spring is in equilibrium when 2 F = 0 = mg — kx0, so the spring stretches an extra amount xQ = m g/k to be in equilibrium. If x is measured from this new equilibrium position, the Hooke's law can be used directly with the s…
Which of the following represents a simple harmonic oscillator: (a) F = -0.5x2 (b) F = -2.3x (c) F = 8.6x (d) F = -4x?
(a) F = -0.5x2 (b) F = -2.3x (c) F = 8.6x (d) F = -4x? Any oscillating system for which the net restoring force is directly proportional to the negative of the displacement (F = -kx) is said to exhibit simple harmonic motion (SHM). Such a system is often called a simple harmonic oscillator (SHO).
Show how am = -kx is satisfied by the equation x = A * cos(ω * t + φ). F = amF = -kxm - object massk - spring constant
This solution does indeed work for any t,
x = A * cos(ω * t + φ) What variables are irrelevant to the frequency of the spring system?
A and φ. A and φ can be different every time we use our spring system. A is given by how far we pull the spring initially before letting it go. φ is given by when we start our timer.
x = A * cos(ω * t + φ) What determines the amplitude of motion of the spring system?
Only how far you initially pulled the spring.
x = A * cos(ω * t + φ) What does the phase angle φ determine?
φ is given by when we start our timer. It's equal to how long after (or before) t=0 x=A is reached.
x = A * cos(ω * t + φ) What variable describes the spring system itself (mass, spring constant)?
Angular frequency: ω.
x = A * cos(ω * t + φ) How do we get to formula ωT = 2π?
What should a be in cos(a + φ) to repeat again in some time? cos(2π + φ) = cos(φ) Therefore whatever the t is - the PERIOD is amount of time that enables us to repeat ourselves (and this can be a different value depending on the angular frequency of course). cos(ωT + φ) = cos(2π + φ) We're trying to find the period of this particular graph.We recognize that φ and A always stay the same for this graph (they stem from the initial conditions).We know that sinusoid repeats itself every 2π.
What's the formula between angular frequency VS motion frequency?
ω = 2πf Because ωT = 2π since period T is any number that lets us repeat the x position. Angular frequency: ω [rad/s] Motion frequency: f [1/s]
What determines frequency f and period T of motion?
ONLY mass (m) and spring constant (k). NOT the amplitude! No matter how big or small the amplitude it, frequency of the spring system will remain the same.
By how much should the mass on the end of a spring be changed to halve the frequency of its oscillations? (a) No change; (b) doubled; (c) quadrupled; (d) halved; (e) quartered.
(c) quadrupled
The position of a SHO is given by x = (0.80 m) cos(3.14t — 0.25). The frequency is (a) 3.14 Hz, (b) 1.0 Hz, (c) 0.50 Hz, (d) 9.88 Hz, (e) 19.8 Hz. SHO - simple harmonic oscillator
(c) 0.50 Hz x = A * cos(ω * t + φ)f = ω / 2π
What's the velocity and acceleration of the mass in SHO? What are vmax and amax? SHO - simple harmonic oscillator
The speed reaches its maximum when the oscillating object is passing through its equilibrium point, x = 0.And the speed is zero at points of maximum displacement, x = ± A. Acceleration has its maximum value where x = ±A.Acceleration is zero at x = 0.
A vibrating floor. A large motor in a factory causes the floor to vibrate at a frequency of 10 Hz. The amplitude of the floor’s motion near the motor is about 3.0 mm. Estimate the maximum acceleration of the floor near the motor.
APPROACH Assuming the motion of the floor is roughly SHM we can make an estimate for the maximum acceleration using Eq. 14-9b. NOTE The maximum acceleration is a little over g, so when the floor accelerates down, objects sitting on the floor will actually lose contact momentarily, which will cause noise and serious wear.
A spring stretches 0.150m when a 0.300-kg mass is gently attached to it. The spring is then set up horizontally with the 0.300-kg mass resting on a frictionless table. The mass is pushed so that the spring is compressed 0.100 m from the equilibrium point, and released from rest. Determine: (a) the spring stiffness constant k and angular frequency w; (b) the amplitude of the horizontal oscillation A; (c) the magnitude of the maximum velocity vmax; (d) the magnitude of the maximum acceleration am…
APPROACH When the 0.300-kg mass hangs at rest from the spring as in Fig. 14-3b, we apply Newton’s second law for the vertical forces: 2 F = 0 = mg — k x Q, so k = mg/x0. For the horizontal oscillations, the amplitude is given, and the other quantities can be found from Eqs. 14-4,14-5,14-7, and 14-9. We choose x positive to the right.
What's the formula for potential energy U in a spring?
Does simple pendulum perform SHM? SHM - simple harmonic motion
Yes, for very small angles. Since F is proportional to the sine of 0 and not to 0 itself, the motion is not SHM. However, if 0 is small, then sin 0 is very nearly equal to 0 when the latter is specified in radians.
What's overdamped, underdamped, and critically damped motion here? The amplitude of any real oscillating spring or swinging pendulum slowly decreases in time until the oscillations stop altogether. This is called damped harmonic motion. The damping is generally due to the resistance of air and to internal friction within the oscillating system. The energy that is dissipated to thermal energy is reflected in a decreased amplitude of oscillation.
A - underdamped, system makes several swings before coming to rest. B - critically damped, equilibrium is reached in the shortest time. C - overdamped, represents the situation when the damping is so large that it takes a long time to reach equilibrium. These terms all derive from the use of practical damped systems such as door-closing mechanisms and shock absorbers in a car (Fig. 14-21), which are usually designed to give critical damping. But as they wear out, underdamping occurs: a door sla…
What's the formula for Fdamping in oscillating spring?
Fdamping = - b v v - velocity of the massb - damping constant The damping force depends on the speed of the oscillating object, and opposes the motion.
Forced oscillations: at what frequency should you push on a swing for its amplitude to increase the largest?
At the natural frequency (resonant frequency) of the swing, the frequency the swing would move without our interaction with it. A simple illustration of resonance is pushing a child on a swing. A swing, like any pendulum, has a natural frequency of oscillation that depends on its length I. If you push on the swing at a random frequency, the swing bounces around and reaches no great amplitude. But if you push with a frequency equal to the natural frequency of the swing, the amplitude increases g…
If a particle undergoes SHM with amplitude 0.18m, what is the total distance it travels in one period?
Tall buildings are designed to sway in the wind.In a 100-km/h wind, for example, the top of the 110-story Sears Tower oscillates horizontally with an amplitude of 15cm. The building oscillates at its natural frequency, which has a period of 7.0 s. Assuming SHM, find the maximum horizontal velocity and acceleration experienced by a Sears employee as she sits working at her desk located on the top floor. Compare the maximum acceleration (as a percentage) with the acceleration due to gravity.
Figure shows two examples of SHM, labeled A and B. For each, what is (a) the amplitude (b) the frequency (c) the period (d) Write the equations for both A and B in the form of a sine or cosine.
Is the acceleration of a simple harmonic oscillator ever zero? If so, where?
Yes, acceleration is zero as the mass passes through the equilibrium point. The acceleration of a simple harmonic oscillator is momentarily zero as the mass passes through the equilibrium point. At this point, there is no force on the mass and therefore no acceleration.
Real springs have mass. Will the frequency be larger or smaller than given by the equations for a mass oscillating on the end of an idealized massless spring? Explain.
The true frequency will be smaller. The spring needs to accelerate not only the mass attached to its end, but also its own mass. As a mass on a spring oscillates, potential energy is converted into kinetic energy. The maximum potential energy depends on the displacement of the mass. This maximum potential energy is converted into the maximum kinetic energy, but if the mass being accelerated is larger then the velocity will be smaller for the same amount of energy. A smaller velocity translates…
How could you double the maximum speed of a simple harmonic oscillator (SHO)?
The maximum speed of a simple harmonic oscillator is given by vmax = A * ω The maximum speed can be doubled by doubling the amplitude A.
For a simple harmonic oscillator, when (if ever) are the displacement and velocity vectors in the same direction? When are the displacement and acceleration vectors in the same direction?
The displacement and velocity vectors are in the same direction while the oscillator is moving away from its equilibrium position. The displacement and acceleration vectors are never in the same direction.
Two equal masses are attached to separate identical springs next to one another. One mass is pulled so its spring stretches 20 cm and the other is pulled so its spring stretches only 10 cm. The masses are released simultaneously. Which mass reaches the equilibrium point first?
The two masses reach the equilibrium point simultaneously. The angular frequency is independent of amplitude and will be the same for both systems.
Does a car bounce on its springs faster when it is empty or when it is fully loaded?
Empty. The period of the oscillation of a spring increases with increasing mass, so when the car is empty the period of the harmonic motion of the springs will be shorter, and the car will bounce faster.
Why can you make water slosh back and forth in a pan only if you shake the pan at a certain frequency?
If you shake the pan at a resonant frequency, standing waves will be set up in the water and it will slosh back and forth. Shaking the pan at other frequencies will not create large waves. The individual water molecules will move but not in a coherent way.
A 5.0-kg trout is attached to the hook of a vertical spring scale, and then is released. Describe the scale reading as a function of time.
This is a case of underdamped motion. Before the trout is released, the scale reading is zero. When the trout is released, it will fall downward, stretching the spring to beyond its equilibrium point so that the scale reads something over 5 kg. Then the spring force will pull the trout back up, again to a point beyond the equilibrium point, so that the scale will read something less than 5 kg. The spring will undergo damped oscillations about equilibrium and eventually come to rest at equilibri…
What's the source of any wave?
Vibration. See the wave pulse for a contrast: A single pulse can be formed on a cord by a quick up-and-down motion of the hand, Fig. 15-2. The hand pulls up on one end of the cord. Because the end section is attached to adjacent sections, these also feel an upward force and they too begin to move upward. As each succeeding section of cord moves upward, the wave crest moves outward along the cord. Meanwhile, the end section of cord has been returned to its original position by the hand. As each…
What the high VS low points are called on a wave?
High: crest Low: trough
Give an example (sea) of why speed of the wave is equal to wavelength * frequency. For example, suppose a wave has a wavelength of 5 m and a frequency of 3 Hz.
Since three crests pass a given point per second, and the crests are 5m apart, the first crest (or any other part of the wave) must travel a distance of 15 m during the 1s. So the wave velocity is 15m/s.
You notice a water wave pass by the end of a pier with about 0.5 s between crests. Therefore (a) the frequency is 0.5 Hz (b) the velocity is 0.5 m/s (c) the wavelength is 0.5 m (d) the period is 0.5 s
(d) the period is 0.5 s
Transverse VS longitudinal waves?
Suppose a drum plays, how exactly does the longitudinal wave occur?
Production of a sound wave, which is longitudinal, shown at two moments in time about a half period (1/2 T) apart.
What's velocity of transverse waves VS velocity of longitudinal waves?
For both: FT - tension in the cordμ - the Greek letter mu, where here μ = m/l (mass/length) E - elastic modulus of the materialρ - material density B - bulk modulusρ - material density