JEE Mains · Physics · STD 11 - Work,energy,power and collision
- JEE Mains 2025
- Hard
- MCQ
A bead of mass ' \(m\) ' slides without friction on the wall of a vertical circular hoop of radius ' \(R\) ' as shown in figure. The bead moves under the combined action of gravity and a massless spring ( k ) attached to the bottom of the hoop. The equilibrium length of the spring is ' \(R\) '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' R ', would be (spring constant is ' k ', g is accleration due to gravity)
Pick an option to check your answer.
Answer & Solution
Correct Answer
(A) \(\sqrt{3 \mathrm{Rg}+\frac{\mathrm{kR}^2}{\mathrm{~m}}}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} \text { Work done by gravity } & =m g\left(2 R-R \cos 60^{\circ}\right) \\ & =\frac{3 m g R}{2}\end{aligned}\)
\(\begin{aligned} \text { Work done by spring } & =-\frac{1}{2} k\left(0^2-R^2\right) \\ & =\frac{1}{2} k R^2\end{aligned}\)
Net work = change in kinetic energy
i.e. \(\frac{3 m g R}{2}+\frac{k R^2}{2}=\frac{1}{2} m v^2\)
or \(\quad v^2=3 g R+\frac{k R^2}{m}\)
or \(\quad v=\sqrt{3 g R+\frac{k R^2}{m}}\)
More questions from Physics
- A transverse wave on a string is described by \(y = 3\sin(36t + 0.018x + \pi/4)\), where \(x, y\) are in cm and \(t\) in seconds. The least distance between the two successive crests in the wave is _____ cm. (Nearest integer)
(\(\pi = 3.14\))JEE Mains 2026 Medium - Which one of the following is the correct dimensional formula for the capacitance in F ? \(\mathrm{M}, \mathrm{L}, \mathrm{T}\) and C stand for unit of mass, length, time and charge,JEE Mains 2025 Easy
- The specific heats, \(C_P\) and \(C_V\) of a gas of diatomic molecules, \(A\), are given (in units of \(J\, mol^{-1}\, K^{-1}\)) by \(29\) and \(22\), respectively. Another gas of diatomic molecules \(B\), has the corresponding values \(30\) and \(21\). If they are treated as ideal gases, thenJEE Mains 2019 Hard
- A spherical body of mass \(100 \mathrm{~g}\) is dropped from a height of \(10 \mathrm{~m}\) from the ground. After hitting the ground, the body rebounds to a height of \(5 \mathrm{~m}\). The impulse of force imparted by the ground to the body is given by : _______. (given \(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\) )JEE Mains 2024 Hard
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is __________ m/s.JEE Mains 2026 Medium
- Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\) : The photoelectric effect does not take place, if the energy of the incident radiation is less than the work function of a metal. Reason \(R\) : Kinetic energy of the photoelectrons is zero, if the energy of the incident radiation is equal to the work function of a metal. In the light of the above statements, choose the most appropriate answer from the options given belowJEE Mains 2022 Medium
More PYQs from JEE Mains
- In an experiment to measure the focal length ( \(f\) ) of a convex lens, the magnitude of object distance \((x)\) and the image distance \((y)\) are measured with reference to the focal point of the lens. The \(y-x\) plot is shown in figure. The focal length of the lens is _______ \(\mathrm{cm}\).
JEE Mains 2024 Hard - In a reflecting telescope, a secondary mirror is used to:JEE Mains 2023 Medium
- A small uncharged conducting sphere is placed in contact with an identical sphere but having \(4 \times 10^{-8} \mathrm{C}\) charge and then removed to a distance such that the force of repulsion between them is \(9 \times 10^{-3} \mathrm{~N}\). The distance between them is (Take \(\frac{1}{4 \pi \epsilon_{\mathrm{o}}}\) as \(9 \times 10^9 \mathrm{in} \mathrm{SI}\) units)JEE Mains 2025 Hard
- In the following \(p-V\) diagram the equation of state along the curved path is given by \((V-2)^2=4\) ap where \(a\) is a constant. The total work done in the closed path is
JEE Mains 2026 Medium - A wire of length \(L\) and mass per unit length \(6.0\times 10^{-3}\; \mathrm{kgm}^{-1}\) is put under tension of \(540\; \mathrm{N}\). Two consecutive frequencies that it resonates at are : \(420\; \mathrm{Hz}\) and \(490 \;\mathrm{Hz}\). Then \(\mathrm{L}\) in meters isJEE Mains 2020 Easy