A particle of mass m = 27 g 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 30 cm (Fig. 7-23). The curved sides of the bowl are frictionless, and for the flat bottom the coefficient of kinetic frictiok = 0.19. The particle is released from rest at the rim, which is 15 cm above the flat part of the bowl.

(c) Where does the particle finally come to rest?
_________cm from point a

Please help...

Yea

To determine where the particle finally comes to rest, we need to analyze the forces acting on it and the resulting motion.

First, let's consider the forces acting on the particle. The only force that acts in the vertical direction is the weight of the particle (mg), where m is the mass of the particle and g is the acceleration due to gravity. The weight acts downward.

In the horizontal direction, two forces are acting on the particle: the normal force (N) and the kinetic friction force (fk). The normal force acts perpendicular to the flat part of the bowl and counteracts the weight of the particle. The kinetic friction force opposes the motion of the particle on the flat surface.

Since the curved sides of the bowl are assumed to be frictionless, the only place where the particle experiences horizontal forces is on the flat bottom part. Therefore, we will consider the motion of the particle only on the flat surface.

At the beginning, when the particle is released from rest, its only horizontal force is the friction force. This force will accelerate the particle until its kinetic friction force becomes equal to the maximum friction force it can exert (μN), where μ is the coefficient of kinetic friction and N is the normal force. Once the friction force reaches its maximum value, the particle will stop accelerating and move at a constant velocity, ultimately coming to rest.

To find where the particle finally comes to rest, we need to calculate the distance the particle travels on the flat surface. We know the coefficient of kinetic friction (μ = 0.19) and the length of the flat surface (30 cm or 0.3 m). We can use the following equation of motion:

fk = μN

Since N is equal to the weight of the particle, N = mg, the equation becomes:

fk = μmg

We also know that the maximum friction force (fk) is given by:

fk = μN

Plugging in the values, we have:

μmg = μN

Since N = mg, we can simplify the equation to:

μmg = μmg

This equation tells us that the maximum friction force (μmg) is equal to itself, which is always true.

After the particle stops accelerating on the flat surface, it will come to rest. At this point, there is no horizontal force acting on it, so the particle will stay in equilibrium. This means that the normal force (N) is equal to the weight of the particle (mg).

To find the final resting position of the particle, we can use the equation of motion:

fk = μN

Since N = mg, the equation becomes:

fk = μmg

At rest, the friction force (fk) is given by:

fk = μN

Substituting N = mg, we have:

fk = μmg

And solving for d (the distance the particle travels on the flat surface):

d = coefficient of kinetic friction x weight x length

Plugging in the values, we have:

d = (0.19)(27 g)(9.8 m/s^2)(0.3 m)

Simplifying the units, we get:

d ≈ 0.15 m

So, the particle finally comes to rest approximately 0.15 m from point a.