A particle of mass $m$ is kept on the top of a smooth sphere of radius $\mathrm{R}$. It is given a sharp impulse which imparts it a horizontal speed v. (a) Find the normal force between the sphere and the particle just after the impulse. (b) What should be the minimum value of $v$ for which the particle does not slip on the sphere? (c) Assuming the velocity $v$ to be half the minimum calculated in part, (b) Find the angle made by the radius through the particle with the vertical when it leaves the sphere.
$\mathrm{N}=\mathrm{mg}-\left(\mathrm{mv}^{2}\right) / \mathrm{R}$
(b) When particle moves with maximum velocity then, $\mathrm{mg}=\left(\mathrm{mv}^{2}\right) / \mathrm{R}$
$\mathrm{mg}=\left(\mathrm{mv}^{2}\right) / \mathrm{R}$
$v=\sqrt{g} R$
(c ) From figure,
$\mathrm{v}_{1}=(\sqrt{g R}) / 2 \ldots \ldots 1$
and
$\left(m v_{2}^{2}\right) / R=m g \cos \theta$
$\mathrm{v}_{2}=(\sqrt{g} \mathrm{R} \cos \theta) \ldots \ldots 2$
Also, change in K.E. = Work done
$\frac{\frac{1}{2}}{m_{2}} v_{2}^{2}-{ }^{\frac{1}{2}} m v_{1}=m g R(1-\cos \theta) \ldots \ldots 3$
from equation 2 and 3
$g R \cos \theta=v_{1}^{2}+2 g R(1-\cos \theta)$
$\theta=\cos ^{-1}\left(\frac{3}{4}\right)$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,