How far does each can slide on the table?

[kashiark]

Homework Statement

In a hardware store, paint cans, which weigh 46N each, are trasnported from storage to the back of the paint department by placing them on a 24 degree ramp. The cans slide down the ramp at a constant speed of 3.4 m/s onto a table made of the same material as the ramp. How far does each can slide on the table?

Homework Equations

mgsin(x) = component of gravity in the same dimension as the incline on which the object sits
KE = mv²/2
ΔKE = Fd

The Attempt at a Solution

46sin(24) ≈ 18.7
-(46/9.8)(3.4)²/2 = -18.7d
d ≈ 1.5
My problem is if the friction force were equivalent to the force of gravity down the incline, the the paint cans would never move.

 

[Slip]

The frictional force and the force propelling the cans down the incline must be equal, as the cans are moving at a constantly velocity, so all forces are in equilibrium.

I think we're assuming the cans are originally given some speed to begin with, and aren't placed on the ramp stationary, as the question does say it has a velocity.

 

[kerimek]

I would like to clarify that the calculations provided in the attempt at a solution are incorrect. The correct equations for this scenario would be:

1. The net force acting on the paint can:
Fnet = mgsin(x) - μmgcos(x)

2. Equating this net force to the mass times acceleration (since the can is moving at a constant speed, the acceleration is 0):
mg(sin(x) - μcos(x)) = 0

3. Solving for μ (the coefficient of friction between the ramp and the can):
μ = tan(x)

4. Now, using the formula for work done by a force (W = Fd), we can calculate the distance each can slides on the table:
Fd = μmgcos(x)d
d = μmgcos(x)/F

5. Plugging in the values given in the problem, we get:
d = (tan(24))(46)(9.8)(cos(24))/(46)
d ≈ 1.5m

Therefore, each can will slide approximately 1.5 meters on the table. It is important to note that this is an ideal scenario and does not take into account factors such as air resistance and imperfections in the surface of the ramp and table. These factors may slightly affect the distance the cans slide.