Solving a Frictionless Bug Problem in a Bowl

[Grogerian]

Homework Statement

A bug slides back and forth in a bowl 11cm deep, the bowl is frictionless except a 1.5cm patch at the *flat* bottom coefficient of friction = .61

Homework Equations

W = f*d
r= 11cm
$$\mu=.61$$
stickypatch=1.5cm

The Attempt at a Solution

i can get the first part (its velocity right when it hits the patch [1.47m/s] afterwards I'm stuck i tried using Work = F*d but i get:
W = -.61(mass)(9.81) * .015m

how am i supposed to solve this without the required mass? i tried finding substitutes to put in for mass or to remove mass, but no luck.

 

[tiny-tim]

A bug slides back and forth in a bowl 11cm deep, the bowl is frictionless except a 1.5cm patch at the *flat* bottom coefficient of friction = .61
…
i can get the first part (its velocity right when it hits the patch [1.47m/s] afterwards I'm stuck i tried using Work = F*d but i get:
W = -.61(mass)(9.81) * .015m

how am i supposed to solve this without the required mass? i tried finding substitutes to put in for mass or to remove mass, but no luck.

Hi Grogerian!

I assume you're using the work-energy theorem …

the mass should appear on both sides, so you can just cancel it at the end.

 

[baba89]

As a scientist, my response would be to approach this problem using the principles of Newton's Laws of Motion and the concept of forces. Since the bowl is frictionless except for the 1.5cm patch at the bottom, we can assume that the bug's motion is affected only by gravity and the friction force from the patch.

First, let's define our coordinate system. We can take the bottom of the bowl as the origin and the direction of the bug's motion as the positive x-axis. We can also assume that the bug starts at rest at the top of the bowl.

Next, we can apply Newton's Second Law (F=ma) to the bug's motion. In this case, the only force acting on the bug is the friction force from the patch. We can express this force as F = μN, where μ is the coefficient of friction and N is the normal force. Since the bug is sliding along the bottom of the bowl, the normal force is equal to the bug's weight, mg. Therefore, we can write:

F = μmg

We can also use the equation for work (W = Fd) to relate the friction force to the distance traveled by the bug. In this case, the distance traveled is the length of the patch, 1.5cm. Therefore, we can write:

W = μmgd

Since we are given the initial velocity of the bug, we can use the work-energy theorem to relate the work done by friction to the change in kinetic energy of the bug. This can be expressed as:

W = ΔKE = (1/2)mvf^2 - (1/2)mvi^2

where vf is the final velocity of the bug and vi is the initial velocity (which is 0 in this case).

Now, we can combine all of these equations to solve for the mass of the bug. This can be done by setting the equations for work equal to each other:

μmgd = (1/2)mvf^2

We can then solve for the mass, m, and plug in the given values for μ, g, d, and vf to get:

m = 2μgd/vf^2

Once we have the mass of the bug, we can use it to solve for the acceleration of the bug using Newton's Second Law. We can then use this acceleration to find the time it takes for the bug to reach the bottom of