If T is the time period of an oscillating pendulum, which one of the following statements is NOT correct ?
- (a)The motion repeats after time T only once
- (b)T is the least time after which motion repeats itself
- (c)The motion repeats itself after nT, where n is a positive integer
- (d)T remains the same only for small angular displacements
Correct — A, The motion repeats after time T only once. The word that breaks this statement is 'only once'. A periodic motion is one that repeats itself over and over, endlessly so long as nothing damps it out. The pendulum returns to the same position with the same velocity after T seconds, and again after 2T, and again after 3T. Saying it repeats after T only once contradicts the very meaning of a time period, so this is the statement that is not correct — which is what the item asks for.
- (b)T is the least time after which motion repeats itself — This is the definition of the time period, so it is correct and cannot be the answer. The word 'least' matters — the motion also repeats after 2T and 3T, and the period is the smallest such interval.
- (c)The motion repeats itself after nT, where n is a positive integer — Correct, and it is the very statement that exposes option (a). If the state recurs at T, 2T, 3T and so on, then it plainly does not repeat 'only once'.
- (d)T remains the same only for small angular displacements — Correct as well. The familiar formula for the period of a simple pendulum, two pi times the square root of length over the acceleration due to gravity, is derived by treating the sine of the angle as equal to the angle, which is a good approximation only for small swings. Push the amplitude up and the real period grows slightly, so the period stops being independent of amplitude.
A simple pendulum swinging through a small arc performs simple harmonic motion. Its time period is two pi times the square root of the length divided by the acceleration due to gravity, so it depends on the length of the string and on the local value of gravity — and not at all on the mass of the bob. Periodic motion means the system returns to the same state at equal intervals, and the time period is the shortest such interval.
Two ideas are being tested at once, and the trap in a NOT-correct item is to stop reading as soon as you meet a statement you recognise. Options (b) and (c) are two halves of the same textbook definition, and (d) is the standard caution about the small-angle approximation. Only (a) says something the definition forbids. A useful habit for this pattern is to test each statement for the one word that could poison it — here it is 'only once', in the same way that many pendulum items turn on the word 'mass'. The independence of the period from the bob's mass, incidentally, is the reason a pendulum clock can be regulated by sliding the bob up or down the rod but not by making the bob heavier.
- The period of a simple pendulum is two pi times the square root of the length divided by g.
- The period does not depend on the mass of the bob or, for small swings, on the amplitude.
- The small-angle approximation behind that formula holds for swings of roughly a few degrees; larger swings lengthen the period.
- Frequency is the reciprocal of the time period, so a one-second pendulum has a frequency of one hertz.
- A pendulum about one metre long has a period of roughly two seconds, which is why seconds pendulums are close to that length.
- Answering a NOT-correct item by picking the first statement that looks familiar.
- Believing a heavier bob swings faster — the period is independent of mass.
- Forgetting that the standard period formula assumes small angular displacements.
NDA asks this either as a statement-screening item like this one, or numerically by changing the length and the mass together and asking for the new period.
Consider the following statements: A simple pendulum is set into oscillation. Then I. The acceleration is zero when the bob passes through the mean position. II. In each cycle the bob attains a given velocity twice. III. Both acceleration and velocity of the bob are zero when it reaches its extreme position during its oscillation. IV. The amplitude of oscillation of the simple pendulum decreases with time. Which of these statements are correct?
- (a) I and II
- (b) III and IV
- (c) I, II and IV
- (d) II, III and IV
Answer(c) I, II and IV
The same statement-screening drill on the same oscillation, testing whether a candidate can tell which quantities vanish where and remembering that a real pendulum is damped.
The time period of oscillation of a simple pendulum having length L and mass of the bob m is given as T. If the length of the pendulum is increased to 4L and the mass of the bob is increased to 2m, then which one of the following is the new time period of oscillation?
- (a) T
- (b) 2T
- (c) 4T
- (d) T/2
Answer(b) 2T
The same formula six months later in the other session, set up so that the mass change is a pure decoy.
The time period of a 1 m long pendulum approximates to
- (a) 6 s
- (b) 4 s
- (c) 2 s
- (d) 1 s
Answer(c) 2 s
The numerical face of the same formula, and the figure worth memorising because it lets you sanity-check any pendulum answer.
- practice — not a real PYQ
The length of a simple pendulum is made nine times its original value. Its time period becomes
- (a)nine times
- (b)three times
- (c)one-third
- (d)unchanged
Answer(b) three times — the period varies as the square root of the length, and the square root of nine is three.
- practice — not a real PYQ
A pendulum clock that keeps correct time at sea level is carried to the top of a high mountain. There it will
- (a)run fast
- (b)run slow
- (c)keep the same time
- (d)stop oscillating
Answer(b) run slow — g is smaller at greater height, so the period lengthens and each 'second' takes longer.