Chapter 13: Oscillations About Equilibrium
In 5/2 periods?
-10 amplitudes
How many amplitudes does an object in simple harmonic motion travel through in one period?
-4 amplitudes
What happens to T when k increases?
-T decreases
What happens to T when mass increases?
-T increases
A mass m is suspended from the ceiling of an elevator by a spring of force constant k. When the elevator is at rest, the period of the mass is T. Does the period increase, decrease, or remain the same when the elevator moves upward with constant speed?
-When the elevator moves upward with constant speed, there is no change in the behavior of the mass and the spring -therefore, the period remains the same
What causes a pendulum system to oscillate?
-a "restoring torque"
In general, to double the period of a mass on a given spring, what must the mass be increased by?
-a factor of 4
What is a large spring constant indicative of?
-a greater spring stiffness
Why does a large spring constant decrease the period of an oscillation?
-a large spring constant indicates a stiffer spring -a mass on a stiff spring completes an oscillation in less time than one on a soft, squishy spring
Why not?
-a larger amplitude implies a larger force exerted by the spring (per Hooke's law) -with a greater force acting on it, the mass moves more rapidly -the speed of the mass is increased just enough that it covers the greater distance in precisely the same time
Why does a large mass increase the period of an oscillation?
-a larger mass has a grater inertia -hence, it takes longer for the mass to move back and forth through an oscillation
What is a periodic motion?
-a motion that repeats itself over and over
As the cart moves towards x = -A, what type of force does it experience?
-a negative force which causes it to decelerate and come to rest at x = -A
What is a physical pendulum?
-a pendulum in which the mass is not concentrated at a point, but is instead distributed over a finite volume
What causes a mass and spring system to oscillate?
-a restoring force
When an object is displaced from a position of stable equilibrium, what does it experience?
-a restoring force
What causes a pendulum system to oscillate?
-a restoring torque
What is k a measure of?
-a spring's stiffness
A pendulum system is what version of a mass and spring system?
-an angular version
What is a "small angle" considered to be?
-angles smaller than about π/8 = 22.5 degrees
When does an object on a spring reach maximum velocity?
-as it passes through equilibrium
Why not?
-because acceleration is not constant
Why?
-because the elevator moves upward with constant speed, there is no change in the behavior of the pendulum
Most notably, what is T independent of?
-both mass m and amplitude A
What effect does a large spring constant have on period?
-decreases period
Why doesn't mass appear in the expression for the period of a pendulum?
-for the same reason that different masses free fall with the same acceleration -->in particular, a large mass tends to move more slowly because of its large inertia; on the other hand, the larger a mass the greater the gravitational force acting on it -->these two effects cancel in free fall, as well as in a pendulum
What are the implication sof this in terms of periods on the Moon or in orbit?
-for the same spring + mass system, one would observe the same period on the Moon or in orbit
What two forces act on a pendulum of mass m?
-force of gravity -tension force
What provides the restoring torque?
-gravity
If a mass m is attached to a given spring, its period of oscillation is T. If two such springs are connected end to end and the same mass m is attached, is the resulting period of oscillation greater than, less than, or equal to T?
-greater than T
What special unit has been introduced for the measurement of frequency?
-hertz
Why doesn't amplitude impact period?
-if an object has a greater amplitude, it spends more time further from the spring equilibrium -->therefore, this object has a larger acceleration that makes up for the greater distance from equilibrium
Soldiers on the march are often ordered to break cadence in their step when crossing a bridge. Why is this a good idea?
-if soldiers march in synchrony, the bridge will oscillate with the frequency of their step -if this frequency is near a resonance frequency of the bridge, the amplitude of oscillation could increase to dangerous levels
In what directly does the tension force act?
-in a radial direction
Why not?
-in moving back to equilibrium, the object has gained speed and momentum
What effect does a large mass have on period?
-increases period
As an object on a spring approaches equilibrium, what happens to its acceleration?
-its acceleration decreases
If the period of an oscillation (T) is very small, what do we expect the corresponding frequency to be?
-large
What is the size of this force?
-mgsinθ
Can the constant acceleration equations be applied to harmonic motions?
-no
Does a spring force have a constant size or direction?
-no
Does amplitude impact period?
-no
Does gravity play a role in determining the period of oscillation?
-no
Does the oscillating object then come to rest?
-no
Does the period of a pendulum vary with weight?
-no
If you set something in motion from a distance A from equilibrium, will it ever go further than A?
-no
Is the spring force constant throughout the movement of a mass and spring?
-no
What impact does amplitude have on the period of an oscillation?
-no impact whatsoever!
Does the spring constant for a given spring ever change?
-no, it does not
Is the speed of the cart at this location equal to zero?
-no, it is not
What is a period?
-one complete oscillation
Describe the direction of the force exerted by a spring?
-opposite in direction to its displacement from equilibrium
What dimensions does an oscillation have?
-oscillations are unitless
Oscillation
-repetitive movement of an object around a position of equilibrium
What type of force is this?
-restoring force
A grandfather clock keeps correct time at sea level. If the clock is taken to the top of a nearby mountain, would you expect it to keep correct time, run slow, or run fast?
-run slow
What are the units of a period?
-seconds
What are the two types of harmonic motion?
-simple and non-simple
Why?
-since gravity is weaker on top of a mountain, the period of the pendulum would increase -as a result, the clock would run slow
If the period of oscillation (T) is very large, what do we expect the corresponding frequency to be?
-small
If we increase the stiffness of a spring, what will happen to the amount of energy stored in it?
-the amount of energy stored in the spring will increase
If we take a spring further from equilibrium, what will happen to the amount of energy stored in it?
-the amount of energy stored in the spring will increase
What is the relationship between amplitude (A) and the total range of motion?
-the amplitude is one-half the total range of motion
For small angles θ (measured in radians), what is the sine of θ approximately equal to?
-the angle itself
If a cart is attached to a spring while at equilibrium, what will happen if it is left undisturbed?
-the cart will remain at rest
What does the amplitude (A) of a motion represent?
-the extreme displacement of an object in simple harmonic motion
What does it supply?
-the force needed to keep the mass moving along its circular path
Compare the k value of a spring that is 1/2x in length to the k value of a spring that is 1x in length. (Assuming that the springs are made of the same materials...)
-the k value of the spring that is 1/2x in length is less than the k value of the spring that is 1x in length
Describe Hooke's law in words
-the larger a spring is displaced from equilibrium, the more force it will exert
Why?
-the longer spring has a smaller force constant; hence its period oscillation is greater than T
What is the frequency of an oscillation?
-the number of oscillations per unit of time
Why not?
-the only factors that impact the period are inherent system properties -->amplitude is an initial condition
Is the period of oscillation for the 1/2 spring greater than, less than, or equal to the period of oscillation of the full spring?
-the period of oscillation for the half-spring is less than for the full spring
When it moves upward with constant acceleration?
-the period of the pendulum will decrease
A pendulum of length L is suspended form the ceiling of an elevator. When the elevator is at rest, the period of the pendulum is T. Does the period increase, decrease, or remain the same when the elevator moves upward with constant speed?
-the period remains the same
What type of function is the motion of the cart over time?
-the position of the cart as a function of time can be represented by a sine or cosine function
What is the "k" in Hooke's law?
-the spring constant
Describe a spring when it is at equilibrium length
-the spring is neither stretched nor compressed
What is the net tangential force acting on m equal to?
-the tangential component of its weight
What is the definition of period (T) for a periodic motion?
-the time required for one cycle of a periodic motion
Why?
-the total elongation of the spring with half the number of coils is half the total elongation of the longer spring -->since half the elongation for the same applied force means a greater force constant, the half-spring has a larger value of k--it is stiffer
A grandfather clock has a weight at the bottom of its pendulum that can be moved up or down by turning a small screw. Suppose you have a grandfather clock that runs slow. Should you turn the adjusting screw so as to (a) raise the weight or (b) lower the weight?
-the weight should be raised
What is the relationship between the direction of the spring force and the direction of the acceleration provided by the spring force?
-they are always in the same direction
Why?
-this shortens the period and makes the clock run faster
In general, in what direction does a restoring force always point?
-towards the equilibrium position
In what direction is the restoring force directed?
-towards the equilibrium position
In what direction do they always point?
-towards the spring equilibrium
What is the direction of a spring force?
-towards the spring equilibrium
When an object on a spring is at its maximum distance from equilibrium, what is velocity equal to?
-velocity is equal to zero
When is the motion of a pendulum virtually identical to the motion of a mass on a spring?
-when a pendulum oscillates with small displacements
When is a vertical spring (with mass) in equilibrium?
-when it exerts an upward force equal to the weight of the mass
What are the only circumstances in which this equation is accurate?
-when the angle is kept small
Why?
-when the elevator moves upward with constant acceleration the system behaves as if the acceleration of gravity were larger -therefore we can see from the equation that the period of the pendulum will increase
When the elevator moves upward with constant acceleration?
-when the elevator moves upward with constant acceleration, the system behaves as if the acceleration of gravity were larger -therefore, the spring is stretched by a greater amount (A is greater), but the period of oscillation remains the same
When does a pendulum have a stable equilibrium?
-when the mass is directly below the suspension point
When a mass is on a spring, when does V max occur?
-when the object passes through equilibrium
During pendulum movement, when does PE = zero?
-when the pendulum has a stable equilibrium/is directly below the suspension point
Why?
-when you have a larger mass, you have a larger inertia -therefore, it takes more time to drag the object around
What two things will an increase in amplitude increase?
-will increase maximum speed -will increase maximum acceleration
Does the period of a pendulum vary with length?
-yes
Is energy conserved in a mass and spring system?
-yes
If a cart is released from rest at a location x=A, what is the net force on the cart when it reaches x = 0?
-zero
What is the KE of a spring when it is at Δx maximum?
-zero
What is the PE of a spring when it is at equilibrium?
-zero
When an oscillating object reaches equilibrium, what is the net force acting on the object equal to?
-zero
With what acceleration does the mass proceed through equilibrium with?
-zero
What are the two primary examples of systems that obey simple harmonic motion?
1. mass and spring system 2. pendulum system
If the cart is displaced from equilibrium by a distance x, what force does the spring exert on the cart?
F = -kx
Which is equal to?
F = mgsinθ
What are the units of k?
N/m
What is PE stored in a spring equal to?
PE spring = 1/2kΔxeq.^2
What is Hooke's law?
Spring force = -kΔxeq.
What is the period of a physical pendulum equal to?
T = 2π * SQRT(I/Mgdcom)
What is the period of a pendulum of small amplitude?
T = 2π * SQRT(L/g)
What is the period of a mass on a spring?
T = 2π * SQRT(m/k)
What is the period of a mass on a spring? (equation)
T = 2π * SQRT(m/k)
What is the relationship between W spring and PE spring?
W spring = -PE spring
What is work done by a spring force equal to?
Work spring = -1/2kΔxeq.^2
What are the units of frequency?
cycles/second
What is the definition of frequency?
frequency = 1/T
When a pendulum is at an angle θ with the vertical, what is the equation that describes its exact vertical height?
h = L(1-cosθ)