Further Mechanics Flashcards
What kind of force is required to keep and object moving in a circle at constant speed?
A constant centripetal force
An object moving in a circle at a constant speed is accelerating. True or False, and why?
True - direction is constantly changing so velocity is constantly changing so it is accelerating
What equations can you use to calculate angular speed, w?
w = v / r w = 2 x π x f
What is angular acceleration in terms of angular velocity in the form of an equation?
Ac = w^2 x r
What is angular acceleration in terms of velocity in the form of an equation?
Ac = v^2 / r
What are the equations for centripetal force?
Fc = m x r x w^2 Fc = (m x v^2) / r
What is a radian?
The angle of a circle sector such that the radius, r, is equal to the arc length, and are normally written in terms of pi. 2π = 360º
What are the conditions for simple harmonic motion?
- Acceleration (a) must be proportional to its displacement (-x) from the equilibrium point
- It must act towards the equilibrium point
What is the constant of proportionality linking acceleration and x in SHM?
- Angular velocity squared (w^2)
or - -k/m
What is x as a trig function of t and w in SHM?
x = Acos(wt)
or
x = Asin(wt)
How can you calculate the maximum speed using w and A?
Max speed = w x A
How can you calculate the maximum acceleration using w and A?
Max acceleration = w^2 x A
What is the equation for the time period of a mass - spring simple harmonic system?
T = 2π √(m/k)
What is the equation for the time period of a simple harmonic pendulum?
T = 2π√(l/g)
What is the small angle approximation for sinx?
sinx = x
Valid in radians
What is the small angle approximation for cosx?
cosx = 1 - (x^2/2)
Valid in radians
Describe the graph for potential energy and kinetic energy against displacement, for a SHM system?
- At the mean position, the total energy in simple harmonic motion is purely kinetic and at the extreme position, the total energy in SHM is purely potential energy
- At other positions, kinetic and potential energies are interconvertible and their sum is equal to 1/2 x k x a^2
- The graph is parabolic in nature
Define simple harmonic motion.
The motion of a body which is acted upon by a resultant force whose magnitude is proportional to the distance of the body from a fixed point and whose direction is always towards that point
Define free vibrations.
The frequency a system tends to vibrate at in a free vibration is called the natural frequency
Define forced vibrations.
A driving force causes the system to vibrate at a different frequency. For higher driving frequencies, the phase difference between the driver and the oscillations rises to π radians. For lower frequencies, the oscillations are in phase with the driving force.
What is resonance in a SHM system?
When vibrations caused by a driving force most efficiently transfers energy to the system, the phase difference will be π/2 radians
Define dampened harmonic motion.
When amplitude diminishes with time as a consequence of a small damping force impeding the motion
Define damping.
Occurs when an opposing force dissipates energy to the surroundings
Explain what is critical damping.
Reduces the amplitude to zero in the quickest time
Explain what is overdamping.
When the damping force is too strong and it returns to equilibrium slowly without oscillation
Explain what is underdamping.
When the damping force is too weak and it oscillated with exponentially decreasing amplitude
What happens to a vibration with greater damping?
The amplitude is lower at all frequencies due to greater energy losses from the system. The resonant peak is also broader
What are some of the implications of resonance in real life?
- Soldiers must break stop when crossing bridges
- Vehicles designed for no unwanted vibrations
