Oscillations

Question 1

Hooke’s Law

Answer 1

When a string is compressed or stretched from its natural length, a force is created

• If the string is displaced by x from its natural lenth, the force it exerts in response is given by the equation:
  
  • F_s= -kx
    
    • The proportionality constant, k, is a positive number called the spring (or force) constant that indicates how stiff the spring is
      
      • The stiffer the spring, the greater the value of k
    • The minus sign in Hooke’s law tells us that F_s and x always point in opposite directions
• The spring tries to restore the block attached to the spring to the equilibrium position, which is the position at which the net force on the block is zero
  
  • The spring provides a restoring force

Question 2

Ideal Springs

Answer 2

Springs that obey Hooke’s law are called ideal or linear springs

• Linear springs provide an ideal mechanism for defining simple harmonic motion ( a kind of vibrational motion)

Question 3

Spring Oscillation

Answer 3

• During oscillation, the force on the block is zero when the block is at equilibrium (x= 0)
  
  • F_s= kx; F_s= 0 at equilibrium
  • Acceleration is also equal to zero at x=0 as a= F_s/m
• At the endpoints of the oscillation region, where the block’s displacement. x, is the largest, the restoring force and the magnitude of the acceleration are both at their maximum

Question 4

Oscillation Energy

Answer 4

• Oscillating block’s motion can be described in terms of energy transfers
  
  • A stretched or compressed spring stores elastic potential energy which is transformed into kinetic energy (and back again)
    
    • The shuttling of energy betseen potential and kinetic energy causes the oscillations
• For a spring with spring constant k, the elastic potential energy it possesses- relative to its equilibrium position -is given by the equation:
  
  • U_s= 1/2kx²
    
    • The farther a spring is stretched or compressed the more work that is done, and as a result the more potential energy stored

Question 5

Oscilliation Energy Transfer

Answer 5

• The maximum displacement from equilibrium is called the amplitude of oscillation, or A
  
  • x= x_max or x= A; x= x_-max or x= -A
• By conservation of mechanical energy, the sum K + U_s is a constant
  
  • At x= (+)(-)x_max U_s is maximized, so K must be minimized (K= 0 at endpoints)
    
    • As the block is passing through equilibrium, x= 0, U_s= 0 and K is maximized
• x= -A
  
  • F_s is max
    
    • a is max
  • U_s is max
  • K= 0
  • v= 0
• x= 0
  
  • F_s= 0
    
    • a= 0
  • U_s= 0
  • K is max
  • v is max
• x= A
  
  • F_s is max
    
    • a is max
  • U_s is max
  • K= 0
  • v= 0

Question 6

Kinematics of SHM

Answer 6

• As the block oscillates, it repeats each cycle of oscillation in the same amount of time
  
  • A cycle is a round-trip (E.G. from x= A to x= -A back to x= A)
    
    • The amount of time it takes to complete a cycle is called the period of oscicllations, or T
      
      • If T is short the block is oscillating rapidly, and if T is long the block is oscillating slowly
      • Rate of oscillation can also be described by counting the number of cycles completed during a time interval; the number of cycles that can be completed per unit of time is called frequency
        
        • The frequency of oscillations, f, is expressed in cycles per second
          
          • Once cycle per second is one hertz (Hz)

Question 7

Frequency and Period of Spring-Block Oscillator

Answer 7

T= 1/f and f= 1/T

• One of the defining properties of the spring-block oscillator is that the frequency and period can be determined from the mass of the block and the force constant of the spring
  
  • f= 1/(2π)√(k/m)
  • T= 2π√(m/k)
    
    • A stiff spring (large k) with small mass (small m) would cause rapid oscillation and high frequency; ratio k/m is large (high frequency) ratio m/k is small (short period)
• In simple harmonic motion, both the frequency and the period are independent of the amplitude

Question 8

x-Position of SHM

Answer 8

The position x can be expressed as a sine or consine function in terms of t

x= Acos(2πft + Ø) or x= Asin(2πft + Ø)

• Ø is called the phase and is determined by the intial conditions
  
  • If x= A at t= 0, then choosing the cosine equation would mean Ø= 0
  • if x= 0 at t= 0, then choosing the sine equation would mean Ø= 0

Question 9

Effective Spring Constant

Answer 9

Force constant of a single spring that would produce the same force on the block attatched to two or multiple springs

• F_eff= -(k₁ + k₂)x
  
  • k_eff= k₁ + k₂

Question 10

Spring Oscillation Vertical Motion

Answer 10

• Gravity causes block to move downward to an equilibrium position, in which the spring would not be in its natural length (in contrast to horizontal spring oscillation)
• A block of mass m attached to a vertical spring is allowed to come to rest, stretching the spring a distance d, at this point the block is in equilibrium
  
  • The upward force of the spring is balanced by the downward force of gravity
    
    • kd= mg → d= mg/k
    • When the block is at a distance y below its equilibrium position, the spring is stretched a total distance of d + y, so the upward spring force is equal to k(d + y) while the downward force stays the same
      
      • F_{net on block}= k(d + y) - mg becomes F= ky because kd= mg

Question 11

Vertical Spring Vs. Horizontal Spring

Answer 11

• The resulting force on the block in a vertical spring is F= ky which has the form of Hooke’s law
  
  • Vertical simple harmonic oscillations of the block have the same characteristics as do horizontal oscillations, with the equilibrium, y= 0 not as the spring’s natural length, but at the point where the hanging block is in equilibrium

Question 12

Pendulums

Answer 12

A simple pendulum consists of a weight of mass m (called the bob) attached to a massless rod that swings, without friction, about the vertical equilibrium position

• The restoring force is provided by gravity, the magnitude of the restoring force when the bob is at an angle θ is given by the equation:
  
  • F_restoring= mgsinθ

Question 13

Pendulum Vs. Spring Block Oscillator Differences

Answer 13

• The displacement of the pendulum is measured by the angle that it makes with the vertical, instead of by its linear distance from the equilibrium position in the case of the spring-block oscillator
• Simple harmonic motion results from a restoring force that has a strength that is proportional to the displacement
  
  • The magnitude of the restoring force on a pendulum is mgsingθ which is not proportional to the displacement θ
    
    • The motion of a simple pendulum is not simple harmonic motion
      
      • However if θ is small, then sinθ is approximately θ (measured in radians) with the magnitude of the restoring force being approximately mgθ which is proportional to θ
        
        • If θ_max is small, the motion can be “treated” as a simple harmonic motion
      • Neither frequency nor period depends on the amplitude (the maximum angular displacement, θ_max) in a pendulum which is a characteristic feature of simple harmonic motion
        
        • f and T of pendulum also do not depend on mass

Question 14

Pendulum Vs. Spring Block Oscillator Similarities

Answer 14

• Displacement is zero at the equilibrium position
• At the endpoints of the oscillation region (where θ= (+)(-)θ_max) the restoring force and the tangential acceleration (a_t) have their greatest magnitudes, the speed of the pendulum is zero, and the potential energy is maximized
• As the pendulum passes through the equilibrium position its kinetic energy and speed are maximized

Question 15

Frequency and Period of Pendulum

Answer 15

• If the restoring force is given by mgθ rather than mgsinθ then the frequency and period of the oscillations depend only on the length of the pendulum and the value of the gravitational acceleration
  
  • f= 1/(2π)√(g/L)
  • T= 2π√(L/g)