10 Oscillations Flashcards
Define Simple harmonic motion. State its defining equation and explain what the equation tells us. State two examples of oscillators undergoing SHM.
SHM is defined as the motion of a particle about a fixed point such that its acceleration is proportional to its displacement from the fixed point and is always directed towards the point. (The fixed point is the equilibrium)
Its defining equation is a = -w^2 x and it tells us that the MAGNITUDE of acceleration a is directly proportional to the magnitude of the displacement x and the direction of acceleration (negative sign) is always in the opposite direction to that of the displacement.
Examples of SHM include the mass-spring system and pendulum systems where the mass or the pendulum that is displaced from its equilibrium position will experience a net force/restoring force that tries to restore it to its equilibrium position.
Define a periodic motion.
A periodic motion is one in which an object continually retraces its path at equal time intervals.
Explain how to show that a motion is simple harmonic.
Since acceleration acts towards equilibrium position, resultant force also acts towards equilibrium position. The resultant force is the restoring force that brings the body back to equilibrium position and acts in the opposite direction to displacement hence by Newton’s 2nd Law of Motion,
Fr = ma = mrw^2
Define angular frequency and state its units. Distinguish between w in circular motion and oscillations.
Angular frequency is defined as the rate of change of phase angle of the oscillation and is equal to the product of 2pi and its frequency. Its units are radian per second.
In circular motion, w represents angular velocity which is the rate of change of angular displacement. However, in oscillations, w represents angular frequency which is the rate of change of phase angle of the oscillation
Define frequency, period, displacement and amplitude.
Frequency f is the number of oscillations per unit time (hertz Hz s^-1).
Period T is the time taken to complete one oscillation.
Displacement X is the distance in a specified direction from the equilibrium position.
Amplitude Xo is the magnitude of the maximum displacement of the particle from its equilibrium position.
Relate T with w, T with f and w with f.
T = 2pi/w T = 1/f w = 2pif
State the equations for the (I) displacement-time, (ii) velocity-time and (iii) acceleration-time graph of a particle that begins its oscillation at the (a) maximum displacement X = Xo and a particle that begins its oscillation at (b) equilibrium where displacement X = 0 and the particle is moving towards the right (+ve).
(a) Beginning at maximum displacement
(i) X = Xocoswt
(ii) V = -wXosinwt
(iii) a = -w^2Xocoswt
(b) Beginning at equilibrium
(i) X = Xosinwt (Note: There should be a negative sign if the particle is first moving towards the left and right is taken to be the positive direction)
(ii) V = wXocoswt
(iii) a = -w^2sinwt
Show how velocity varies with displacement. Draw the graph and state the important features.
v = +/-w √(Xo^2 - X^2)
Elliptical shape
V = 0 at x = Xo
Vmax at x = 0
Show how acceleration varies with displacement. Draw the graph.
a = -w^2x
Straight-line that passes through the origin with a constant negative gradient of magnitude w^2.
State the two equations that govern the motion of an oscillating mass-spring system (both horizontal and vertical) and explain what they mean.
w = √(k/m)
T = 2π√m/k
The motion of an oscillating mass-spring system (both horizontal and vertical) depends only on the mass m and spring constant k.
State the two equations that govern the motion of a simple pendulum system and explain what they mean.
w = √(g/L)
T =2π√(L/g)
The motion of a simple pendulum system depends only on the length L and gravitational acceleration g.
Explain what a free oscillation is.
If an object is displaced from its equilibrium position and then released, it oscillates at its natural frequency about the equilibrium position. A free oscillation occurs when an object oscillates with no resistive and driving forces acting on it and its total energy and amplitude remains constant with time.
For a simple harmonic oscillator, where the system does no work against dissipative forces, its total energy remains constant with time and there is a continual change of kinetic energy to potential energy and vice-versa during the cycle, state the formula to calculate the total energy of the system at any instant during the motion.
Assuming that there are no dissipative forces and total energy remains constant with constant amplitude,
Total energy = Kinetic energy + Potential energy
Kinetic energy
= 1/2 mv^2
= 1/2 m(+/-√(Xo^2 - X^2))^2
= 1/2 mw^2(Xo^2 - X^2)
When X = 0, Total energy = Maximum kinetic energy = 1/2 mw^2Xo^2 where Xo is the amplitude of the motion
Potential energy
= 1/2 kx^2 (since w = √(k/m), k = mw^2)
= 1/2 mw^2x^2
When X = Xo, Total energy = Maximum potential energy = 1/2 mw^2Xo^2
State the equations to show how kinetic energy and potential energy vary with time. Show that Total energy E is constant with time.
Suppose X = Xocoswt and V = -wXosinwt
Kinetic energy = 1/2 mv^2 = 1/2 mW^2Xo^2(sinwt)^2
Potential energy = 1/2 kx^2 = 1/2 kXo^2(coswt)^2 = 1/2 mw^2Xo^2(coswt)^2
Total energy = 1/2 mW^2Xo^2(sinwt)^2 + 1/2 mw^2Xo^2(coswt)^2 = 1/2 mW^2Xo^2
Recall that cos^2X + sin^2X = 1
What is the relationship between the Displacement-time and Energy-time graphs?
The frequency of the energy variation graph is twice that of the motion.
