Question

A particle of mass 1 g executes an oscillatory motion on the concave surface of a spherical dish of radius 2m placed on a horizontal plane. If the motion of the particle beings from a point on the dish at a height of 1 cm from the horizontal plane, the total distance covered by the, particle before it comes to rest, is curve surface is smooth and u = 0.01 for horizontal surface.

• A. 2 m
• B. 10 m
• C. 1 m
• D. 20 m

Answer 1

Answer: (C)

1 m

Answer 2

The problem describes a particle oscillating on a smooth concave spherical dish. It states that the "curve surface is smooth" but "u = 0.01 for horizontal surface". The particle starts at a height h = 1 cm from the lowest point and eventually comes to rest due to friction. We need to find the total distance covered by the particle.

1. Identify Initial Energy: The particle starts from rest at a height h above the lowest point. Its initial mechanical energy is entirely potential energy relative to the lowest point. Initial Potential Energy, PE_initial = mgh where m is the mass of the particle, g is the acceleration due to gravity, and h is the initial height.

2. Identify Final Energy: The particle comes to rest at the lowest point of the dish. At this point, its kinetic energy is zero, and its potential energy (relative to the lowest point) is also zero. Final Energy, E_final = 0

3. Energy Dissipation: The difference between the initial and final energy is dissipated as work done against friction. Energy dissipated, ΔE = PE_initial - E_final = mgh

4. Work Done by Friction: The problem states that the curved surface is smooth, meaning no friction on the curved part. However, it specifies μ = 0.01 for a "horizontal surface". This implies that the energy loss occurs due to friction when the particle moves on an effectively horizontal surface, which is the bottom of the dish where the tangent is horizontal. The work done by friction is given by W_friction = F_friction × d_total, where F_friction is the friction force and d_total is the total distance covered. For a horizontal surface, the normal force N = mg. Therefore, the friction force F_friction = μN = μmg.

5. Equate Energy Dissipation and Work Done by Friction: By the work-energy theorem (or conservation of energy with non-conservative forces), the energy dissipated equals the work done by friction: mgh = μmgd_total

6. Calculate Total Distance: From the equation above, we can solve for d_total: d_total = h / μ

Now, substitute the given values: h = 1 cm = 0.01 m μ = 0.01

d_total = 0.01 m / 0.01 d_total = 1 m

The mass of the particle (1 g) and the radius of the spherical dish (2 m) are extra information not required for this calculation, assuming the friction acts effectively as μmg over the total distance.