Exams › JEE Advanced › Physics
A particle moves in a horizontal circle on the smooth inner surface of a hemispherical bowl of radius R. The plane of circular motion is at a depth d below the centre of the hemisphere. Find the speed of the particle.
- sqrt(g*(R² - d²) / R)
- sqrt(g*(R² - d²) / d)
- sqrt(g*R / (R² - d²))
- sqrt(g*d² / (R² - d²))
Correct answer: sqrt(g*(R² - d²) / d)
Solution
The particle is on the bowl at depth d below the centre. The position vector from the bowl's centre to the particle has magnitude R. The vertical component of the position is d (downward), so the normal N points from particle toward the sphere centre. The angle theta from the vertical satisfies cos(theta) = d/R. Vertical: N*cos(theta) = mg => N*(d/R) = mg => N = mgR/d. Radius of circular path: r = R*sin(theta) = sqrt(R²-d²). Horizontal (centripetal): N*sin(theta) = mv²/r => (mgR/d)*(sqrt(R²-d²)/R) = mv²/sqrt(R²-d²). => g*sqrt(R²-d²)/d = v²/sqrt(R²-d²). => v² = g*(R²-d²)/d. => v = sqrt(g*(R²-d²)/d).
Related JEE Advanced Physics questions
- A particle with mass m moves in the xy-plane, where its velocity at the coordinates (x, y) is described by v = α(yx̂ + 2xŷ), with α being a constant that is not zero. What is the force F experienced by the particle?
- A 5 kg block is moving along the x-axis under the influence of a force given by F = (−20x + 10) N, where x is in meters. At t = 0 seconds, the block is initially at rest at x = 1 meter. What are the block's position and momentum at t = (π/4) seconds?
- A particle strikes a smooth inclined plane horizontally and bounces back in the vertically upward direction. If the incline makes an angle of 60 deg with the horizontal, find the coefficient of restitution between the particle and the inclined surface.
- Ball P has mass 1 kg and is moving at 21 m/s. Ball Q has mass 2 kg and is moving at 4 m/s directly toward P. They undergo a head-on collision. After the collision, P continues in its original direction at 1 m/s. Which of the following statements is/are correct? (I) The speed of Q after the collision is 6 m/s, directed opposite to its initial direction of motion. (II) The coefficient of restitution e = 0.2. (III) The kinetic energy lost in the collision is 200 J. (IV) The magnitude of the impulse exchanged between the two balls is 400 N-s.
- Two blocks of masses m1 and m2 are connected by a spring of spring constant k and placed on a plank lying on a frictionless horizontal surface. The coefficient of friction between m1 and the plank is mu1, and between m2 and the plank is mu2. The plank is given a sudden horizontal acceleration of magnitude a. Which of the following statements is correct regarding the motion of the blocks immediately after the plank starts moving?
- A bead of mass m slides on a fixed horizontal circular wire of radius R. At time t=0, the bead is given a speed v0 along the tangent to the wire. The coefficient of kinetic friction between the bead and wire is muₖ. What is the magnitude of the tangential acceleration of the bead at t=0?
⚔️ Practice JEE Advanced Physics free + battle 1v1 →