Q.A person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of 2.0s−1 and an amplitude 5.0 cm. A weighing machine on the platform gives the persons weight against time.
(a) Will there be any change in weight of the body, during the oscillation?
(b) If answer to part
(a) is yes, what will be the maximum and minimum reading in the machine and at which position?
Simple Harmonic Motion: The Natural Rhythm of Things
Imagine a ball placed at the bottom of a perfectly smooth, U-shaped bowl. If you give it a gentle push, what happens? It rolls up one side, slows down, stops for an instant, then rolls back down, past the bottom, up the other side, stops, and returns. Left alone, it keeps doing this — back and forth, back and forth — in a steady, repeating rhythm.
That rhythm is the heart of Simple Harmonic Motion (SHM). It's the most fundamental kind of oscillatory (back-and-forth) motion in physics.
The Intuition: A Restoring Force That Fights Displacement
The key idea is this: the further you push the object from its resting (equilibrium) position, the stronger the force that tries to pull it back.
In the bowl, when the ball is at the bottom (equilibrium), gravity pulls straight down, and the bowl pushes straight up — no sideways force. But when you push the ball up the side, gravity now has a component that pulls it down the slope. The higher up the side you push it, the steeper the slope, and the stronger that pull-back force becomes.
This is a restoring force — it always points toward equilibrium. And crucially, in SHM, this restoring force is directly proportional to the displacement from equilibrium. Double the displacement, double the restoring force.
F=−kx
F is the restoring force.
x is the displacement from equilibrium.
k is a positive constant (the "stiffness" of the system).
The minus sign is crucial: it tells you the force is opposite to the displacement.
The Precise Statement
Simple Harmonic Motion is the motion of an object where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.
That's it. That single condition — F=−kx — is the entire definition. Everything else (the sine waves, the formulas for period and frequency) follows mathematically from this one law.
What Does This Motion Look Like?
If you track the ball's position over time, you get a beautiful, smooth wave — a sine wave (or cosine wave). It's the same shape as the shadow of a spinning wheel cast on a wall.
The motion has three key descriptors:
Amplitude (A): The maximum displacement from equilibrium. How far you initially pushed the ball up the side of the bowl.
Period (T): The time it takes to complete one full back-and-forth cycle (e.g., from the leftmost point, back to the leftmost point).
Frequency (f): How many cycles happen per second. f=1/T.
Note
A remarkable fact: for a given system (fixed k and fixed mass m), the period and frequency do not depend on the amplitude. A big push and a tiny push take exactly the same time to complete one cycle. This is called isochronism — and it's why pendulums were used to keep time in clocks.
The Mathematical Description (Derived from F=−kx)
Using Newton's second law (F=ma) and the definition of acceleration (a=dt2d2x), the condition F=−kx becomes:
mdt2d2x=−kx
This is a differential equation. Its solution — the position as a function of time — is:
x(t)=Acos(ωt+ϕ)
Where:
ω=mk is the angular frequency (radians per second). It tells you how fast the oscillation is.
ϕ is the phase constant (determines where in the cycle you start measuring time). …
When the platform oscillates vertically, it has an acceleration a=−ω2y, where y is displacement from equilibrium. The weighing machine measures the normal force, which equals the person's apparent weight. At different positions, the platform's acceleration changes the normal force.
Step 1: The angular frequency is ω=2πf=2π×2.0=4πrad/s. Maximum acceleration occurs at extreme positions: amax=ω2A=(4π)2×0.05=16π2×0.05=0.8π2≈7.9m/s2.
Step 2: At the lowest point (maximum downward displacement), acceleration is upward. Applying Newton's second law:
A weighing machine reads the normal force, which changes with the platform's acceleration in SHM. With amax=ω2A≈7.9m/s2, the reading is maximum ≈90.3 kg at the lowest point and minimum ≈9.7 kg at the highest point.
(a) Does the reading change?
Yes. The machine measures the normal force N it exerts on the person, not the true weight mg. In SHM the platform accelerates as a=−ω2x, so N must supply the net force ma and therefore varies through the cycle.
(b) Maximum and minimum readings
Angular frequency. With f=2.0s−1,
ω=2πf=4πrad/s,ω2=16π2≈158rad2/s2.
Maximum acceleration. With amplitude A=5.0cm=0.05m,
amax=ω2A=158×0.05≈7.9m/s2.
Newton's second law (upward positive): N−mg=ma, so N=m(g+a). The machine reading in kilograms is N/g.
Lowest point — platform accelerates upward, a=+amax: …
Step 1: Angular frequency from the given oscillation frequency: ω=2πf=2π(2.0)=4πrad/s.
Step 2: Maximum acceleration of the platform occurs at the extreme positions: amax=ω2A=(4π)2(0.05)≈7.9m/s2.
Step 3: The weighing machine reads the normal force N, which satisfies N−mg=ma (taking up as positive). At the lowest point the platform's acceleration points upward (a=+amax), so Nmax=m(g+amax)=50(9.8+7.9)≈885N≈90.3kg-wt. …
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
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Q.Which of the following function of time represent periodic motion?
(a) sin ωt + cos ωt
(b) e^(-iωt)
(c) e^(iωt)
(d) log ωt
›Reveal solutionSolution
sin ωt + cos ωt represents periodic motion; the others do not represent physically valid periodic motion.
A function f(t) represents periodic motion if it is a real, bounded, physically meaningful function of time satisfying f(t + T) = f(t) for some fixed T.
(a) sin ωt + cos ωt: this is a real, bounded function; it can be written as √2 sin(ωt + π/4), which clearly repeats with period T = 2π/ω. This is genuine periodic motion (in fact, SHM).
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Q.Assertion (A): A uniform circle motion is a simple harmonic motion. Reason (R): The projection of uniform circular motion on any diameter of a circle of reference is simple harmonic motion.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is also false.
›Reveal solutionSolution
The projection of uniform circular motion on any diameter is genuinely simple harmonic motion (Reason is a true, standard NCERT fact) and is exactly the basis for the well-known link between circular motion and SHM stated in the Assertion.
Reason (R): "The projection of uniform circular motion on any diameter of a circle of reference is simple harmonic motion" — this is a precisely correct, standard NCERT result. If a particle P moves with constant angular speed ω around a circle of radius A, the foot of the perpendicular from P onto any diameter (its projection) oscillates back and forth with displacement x(t)=Acos(ωt+ϕ), which is exactly the equation of SHM.
Assertion (A): as commonly used in Class 11 question banks, this statement is intended to express the classic textbook connection between uniform circular motion and SHM (that SHM can be understood/derived via uniform circular motion) — with R giving the precise mechanism (the projection) that establishes this connection.
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Q.What is the minimum condition for a system to execute S.H.M.?
›Reveal solutionSolution
A system executes SHM only if the force pulling it back to equilibrium is proportional to displacement and always points toward the mean position: F=−kx.
Simple harmonic motion (SHM) is a special type of periodic oscillatory motion. For a system to execute SHM, there must exist a restoring force (or, equivalently, a restoring torque for angular SHM) that satisfies two conditions simultaneously:
The magnitude of the restoring force must be directly proportional to the magnitude of the displacement x of the body from its mean (equilibrium) position: F∝x.
The restoring force must always act in the direction opposite to the displacement, i.e., it always tries to bring the body back toward the mean position.
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