A uniform rod of length $L$ lies on a smooth horizontal table. A particle moving on a table strikes the rod perpendicularly at an end and stops. Find the distance travelled by the centre of the rod by the time it turns through a right angle. Show that if the mass of the rod is four times that of the particle, the collision is elastic.
Momentum conservation
$P_{i}=P_{f}$
$m u+0=m(0)+M(v)$
$\mathrm{V}=\frac{\operatorname{mu}}{M}$
Angular Momentum conservation $L_{i}=L_{f}$
$0+m u\left(\frac{\Sigma}{2}\right)=\left(\frac{M L^{2}}{12}\right) \omega+0$
$\omega=\frac{6 m u}{M L}$
Time to rotate by $90^{\circ}$
$\omega=\frac{\theta}{t}$
$t=\frac{(\pi / 2)}{\left(\frac{6 m u}{M L}\right)}$
$\mathrm{t}=\frac{\pi M L}{12 \mathrm{mu}}$
distance travelled by rod
distance $=$ speed $x$ time
$=\left(\frac{m U}{M}\right) \times\left(\frac{\pi M L}{12 m u}\right)$
$\mathrm{S}=\frac{\pi L}{12}$
Now, if $\mathrm{M}=4 \mathrm{~m}$ then
$K E_{i}=\frac{1}{2} m u^{2}$
$K E_{f}=\frac{1}{2} M v^{2}+\frac{1}{2} I \omega^{2}$
$\frac{1}{2}(4 m)\left(\frac{m u}{4 m}\right)^{2}+\frac{1}{2}\left(\frac{M L^{2}}{12}\right)\left(\frac{6 m u}{M L}\right)^{2}$
$K E_{f}=\frac{1}{2} m u^{2}$
Hence proved.
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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,