physics distance due to friction?

Question

A 3.3 kg block starts (with its center-of-mass
vertical displacement at a height of 39 cm) on
a plane that has an inclination angle of 30degrees as
in figure.
Upon reaching the bottom, the block slides
along a horizontal surface. The coeffcient of
friction on both surfaces is meu= 0.3

How far does the block slide on the hori-
zontal surface before coming to rest? Answer
in units of m.

Answer 1

I am surprised that no one answered this question yet. It's not that difficult:

While the block is on the incline, there is a net force of gravity and friction to move the block. Once the block is off the incline, the friction force will bring the block to the stop, so the problem has parts.

On the incline,
F(net force on incline) = F(gravitation pull along the incline) - F(friction on the incline) = F(gravity) x sin 30° - meu x N (normal force on the incline, or the gravity pull perpendicular to the incline) = m x g x sin 30° - meu x m x g x cos 30 ° = m x a(along the incline)

Since we are looking for a(along the incline) only:
a(along the incline) = 9.81 x ( sin 30° - 0.3 x cos 30°) = 2.36 m/sec^2

We then need to look for the velocity of the block at the bottom of the incline:
V^2 = Vo^2 + 2 a(along the incle) x s (distance traveled or the length of the incline) = 0 + 2 x 2.36 x ( 0.39 / sin 30°) = 3.68
V = 1.917 m/sec

The acceleration due to the fricitional force on the horizontal surface is then:
a(due to friction on surface) = F (friction on surface) / m = meu x Fg / m = 0.3 x m x g / m = 0.3 x 9.81 = 2.94 m / sec^2

Using the same formula:
0 = V^2 + 2 a(due friction on surface) x s(distance travel on surface)
0 = 1.917^2 - 2 x 2.94 x s
s = 0.625 m ---------------- -----------------> Answer

You probably have to chew on my solution for a while. It would take me under 5 minuites to do this problem, but took me more than 15 min. to type this. Make a few shortcuts and kinda losing patience with it. Also, from the look of it, we don't even need to know what the mass of the block is, which would make the problem just that much more difficult. :-) Good luck.


XR