Question

8. A particle of mass m = 30 slides inside a bowl whose cross section has circular arcs at each side and a flat horizontal central portion between points a and b of length 20 cm (Fig. 7-3). The curved sides of the bowl are frictionless, and for the flat bottom the coefficient of kinetic friction mu_{k} = 0.21 The particle is released from rest at the rim, which is 10 cm above the that part of the bowl. (a) What is the speed of the particle at a deg b) What is the speed of the particle at b? (c) Where does the particle finally come to rest?

Answer 1

(a) The speed of the particle at point a is approximately 2.42 m/s.

(b) The speed of the particle at point b is approximately 2.87 m/s.

(c) The particle finally comes to rest 10 cm from the bottom of the bowl.

(a) To determine the speed of the particle at point a, the conservation of energy principle is applied. The initial potential energy at the rim is converted into kinetic energy, giving a speed of approximately 2.42 m/s.

(b) For the speed at point b, the conservation of energy is again employed, taking into account the potential energy loss and the work done against friction. The speed at point b is found to be approximately 2.87 m/s.

(c) The particle comes to rest when its kinetic energy is fully dissipated due to friction. The stopping point is determined to be 10 cm above the bottom of the bowl, where the remaining potential energy equals the work done against friction. This scenario considers the interplay between kinetic and potential energy, along with frictional forces, providing insight into the particle's motion within the bowl.