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Thursday, April 17, 2008

AP Physics B & C – Oscillations and Simple Harmonic Motion – Equations to be Remembered

You must remember the following points to make you strong in answering multiple choice questions involving oscillations and simple harmonic motion :
(1) The simplest equation of simple harmonic motion is
y = Asinωt if initial phase and displacement are zero. Here ‘y’ is the displacement, ‘ω’ is the angular frequency and A is the amplitude.
y = Acosωt also represents simple harmonic motion but it has a phase lead of π/2 compared to the above one.
If there is an initial phase of Φ the equation is
y = Asin(ωt + Φ).
[Or, y = Acos(ωt + Φ), if you use the cosine form]
y = Asinωt + Bcosωt represents the general simple harmonic motion of amplitude √(A2 + B2) and initial phase tan-1(B/A).
The above equation can be modified by putting
A = A0cos Φ and
B = A0sin Φ to yield
y = A0cos Φ sinωt + A0sin Φ cosωt = A0sin(ωt + Φ) which is the standard equation of a simple harmonic motion. [Evidently, A2+B2 = A02 and B/A = tan Φ].
(2) The differential equation of simple harmonic motion is
d2y/dt2 = -ω2y
Note that ω =√(k/m) where ‘k’ is the force constant (force per unit displacement) and ‘m’ is the mass of the particle executing the SHM.
(3) Velocity of the particle in SHM, v = ω√(A2 – y2)
Maximum velocity, vmax = ωA
(4) Acceleration of the particle in SHM, a = - ω2y
Maximum acceleration, amax = ω2A
(5) Kinetic Energy of the particle in SHM, K.E. = ½ m ω2( A2 –y2)
Maximum Kinetic energy = ½ m ω2A2
Potential Energy of the particle in SHM, P.E. = ½ m ω2y2
Maximum Potential Energy = ½ m ω2A2
Total Energy in any position = ½ m ω2A2
Note that the kinetic energy is maximum in the mean position and the potential energy is maximum in the extreme position. The sum of the kinetic and potential energies which is the total energy is a constant in all positions. Remember this:
Maximum K.E. = Maximum P.E. = Total Energy = ½ m ω2A2
(6) Period of SHM = 2π√(Inertia factor/ Spring factor)
In cases of linear motion as in the case of a spring-mass system or a simple pendulum, period, T = 2π√(m/k) where ‘m’ is the mass and ‘k’ is the force per unit displacement.
In the case of angular motion, as in the case of a torsion pendulum,
T = 2π√(I/c) where I is the moment of inertia and ‘c’ is the torque (couple) per unit angular displacement.
You may encounter questions requiring calculation of the period of seemingly difficult simple harmonic oscillators. Understand that the question will become simple once you are able to find out the force constant in linear motion and torque constant in angular motion. You will usually encounter cases of linear simple harmonic motion and it won’t be difficult to find he force constant and the period.
(7) In the case of the oscillations of a mass m on a spring of negligible mass, the inertia factor is the mass m attached to the spring and the spring factor is the force constant (spring constant) k of the spring so that the period of oscillation is given by
T = 2π√(m/k)
If two springs of spring constants k1 and k2 are connected in series as shown, the effective spring constant k is given by the reciprocal relation,
1/k = 1/k1 + 1/k2 so that k = k1k2/(k1+k2)
[If many springs are connected in series, you will write 1/k = 1/k1 + 1/k2 +1/k3 + ……etc.]
If springs are connected in parallel as shown in the figure, the effective spring constant will be the sum of the individual spring constants.

If springs are connected on opposite sides of a mass as shown, again the effective spring constant is the sum of the individual spring constants.
If two masses (m1 and m2) are connected by a spring of force constant k and the system is placed on a smooth surface, on compressing the spring by pushing the masses towards each other simultaneously and releasing, the masses oscillate with a period
T = 2π√(m/k) where the effective mass m = m1 m2 /(m1 + m2)
(8) The period of oscillation of a simple pendulum of length is given by
T = 2π√( /g) where g is the acceleration due to gravity.
[Note that the period of oscillation of a spring mass system is independent of the acceleration due to gravity, unlike the simple pendulum].
Questions in this section will be discussed in the next post.
Meanwhile, find many useful multiple choice questions (MCQ) with solution from different branches of physics at Physicsplus

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