Spring-Mass System — Series and parallel combinations
SHM
8
JEE Qs
8%
Hard
60
min
Master the technique of identifying series and parallel combinations by analyzing the force and displacement relationships for each spring upon a small perturbation of the system.
🧮 Key Formulas
✅ Key Points for JEE
- 1**Series Combination**: Springs are in series if they experience the *same restoring force* when the system is displaced, and their *individual elongations/compressions add up* to the total displacement. The effective spring constant (k_eff) for series combination is always less than the smallest individual spring constant.
- 2**Parallel Combination**: Springs are in parallel if they undergo the *same elongation/compression* when the system is displaced, and the *total restoring force is the sum* of individual forces exerted by each spring. The effective spring constant (k_eff) for parallel combination is always greater than the largest individual spring constant.
- 3**Identifying Combinations**: To determine if springs are in series or parallel, imagine displacing the mass by a small amount 'x' and analyze the forces and displacements in each spring. If each spring experiences the same 'x', they are parallel. If each spring experiences the same 'F' (restoring force), they are in series.
- 4Once the effective spring constant (k_eff) for the entire system is determined, the time period of oscillation for a mass 'm' attached to the system can be directly calculated using T = 2π sqrt(m/k_eff).
⚠️ Common Mistakes
- ✕Misidentifying series and parallel arrangements, especially in diagrams where springs are not geometrically aligned in a simple manner (e.g., springs connected via pulleys or inclined planes).
- ✕Algebraic errors when calculating k_eff for series combinations (e.g., forgetting to take the reciprocal after summing 1/k_i).
- ✕Not properly analyzing the connection points and fixed supports. A spring connected between two moving masses often requires a more advanced analysis (reduced mass or system of equations) which is beyond simple series/parallel unless one mass is considered fixed relative to the oscillation of the other.
📝 Practice Questions
See allQ36.A particle oscillates along the x-axis according to the law, x(t) = x0 sin2 ( 2t ) where x0 = 1 m . The kinetic energy (K) of the particle as a function of x is correctly represented by the graph (1) (2) (3) (4)
Q28.Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1 and k2 , respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is (1) k1 (2) k2 √k1k2 (3) (4) k2 k1 √k2k1
Q41.A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is yπ × 10−2 s, where the value of y is (Acceleration due to gravity, g = 10 m/s2 , density of water = 103 kg/m3 ) 2025 (23 Jan Shift 1) JEE Main Previous Year Paper (1) 6 (2) 2 (3) 4 (4) 1
Q36.A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then D is d (1) 16 (2) 10 5 (3) 15 (4) 25 4
Q35.Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Knowing initial position x0 and initial momentum p0 is enough to determine the position and momentum at any time t for a simple harmonic motion with a given angular frequency ω. Reason (R): The amplitude and phase can be expressed in terms of x0 and p0 . In the light of the above statements, choose the correct answer from the options given below : (1) (A) is false but (R) is true (2) (A) is true but (R) is false (3) Both (A) and (R) are true but (R) is NOT the (4) Both (A) and (R) are true and (R) is the correct correct explanation of (A) explanation of (A)
Q24.The displacement of a particle executing SHM is given by x = 10 sin (wt + π3 )m. The time period of motion is 3.14 s. The velocity of the particle at t = 0 is ______ m/s.
NCERT Chapters
- Class 11 Physics Ch 14: Oscillations