A particle of mass m is attached to one end of a massless spring of force constant k , lying on a frictionless horizontal plane. The other end of the spring is fixed. the particle starts moving horizontally from its equilibrium position at time t= 0 with an initial velocity u₀ . When the speed of the particle is 0.5 u_0, it collides elastically with a rigid wall. After this collision,

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

The time at which the particle passes through the equilibrium position for the second time is $t = \frac{5 π}{3} \sqrt{\frac{m}{k}}$

The speed of the particle when it returns to its equilibrium position is u₀ .

The time at which the maximum compression of the spring occurs is $t = \frac{4 π}{3} \sqrt{\frac{m}{k}}$

The time at which the particle passes through the equilibrium position for the first time is $t = π \sqrt{\frac{m}{k}}$

answer is A, D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

$v = {ul}_{0} \cos ω t$ (suppose t₀₁ is the time of collision)

$\frac{u_{0}}{2} = u_{0} \cos {ωt}_{1} \Rightarrow t_{1} = \frac{π}{3 ω}$ Now the particle returns to equilibrium position at time $t_{2} = 2 t_{1}, i.e., \frac{2 π}{3 ω}$ with the same mechanical energy, i.e., its speed will be u₀ . Let t be the time at which the particle passes through the equilibrium position for the second time.
Energy of particle and spring remains conserved.

$t_{3} = \frac{T}{2} + 2 t_{1} = \frac{π}{ω} + \frac{2 π}{3 ω} = \frac{5 π}{3 ω} = \frac{5 π}{3} \sqrt{\frac{m}{k}}$

Watch 3-min video & get full concept clarity

tricks from toppers of Infinity Learn

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET