Concepts Of Physics MCQ Edition [Volume 1]PhysicsWork and Energy
A particle of mass m rests at the top of a smooth sphere of radius R . A sharp impulse is applied, providing the particle with a horizontal speed v . Find the normal force exerted by the sphere on the particle immediately following the impulse. Furthermore, determine the minimum value of v required so that the particle avoids sliding on the sphere. Finally, assuming the initial speed v is exactly half of this minimum
Options
- Amg + mv^2 R , 2Rg , ⁻¹ ( 2 3 )
- Bmg - mv^2 R , 2Rg , ⁻¹ ( 2 3 )
- Cmg - mv^2 R , Rg , ⁻¹ ( 3 4 )
- Dmg - mv^2 R , Rg , ⁻¹ ( 1 2 )
Correct answer
C. mg - mv^2 R , Rg , ⁻¹ ( 3 4 )
Step-by-step solution
At the top of the sphere, the equation of motion along the radial direction is mg - N = mv^2 R . The normal force immediately following the impulse is N = mg - mv^2 R . For the particle to avoid sliding and leave the sphere immediately, the normal force must be zero or less, so N 0 . mg - mv^2 R 0 v Rg . The minimum speed required is v_ min = Rg . Given the initial speed v = 1 2 v_ min = Rg 2 . Let the particle leave the sphere at an angle with the vertical. By conservation of mechanical energy, 1 2 mv^2 + mgR = 1