Lesson 25: Gravity on Inclined Planes

You need to be especially careful when you are doing problems involving gravity pulling something down a slope.

Example 1:Determine the acceleration of a 15 kg box down a 30° slope if the coefficient of friction is 0.15.

The first thing we should do is sketch a free body diagram of the situation.

Figure 1

There are a few special things you have to notice about this diagram:

We can calculate the force due to gravity...

Fg = mg = 15 (9.81) = 147.15 N
(Keep this unrounded number ready, so we can use it in calculations later.)

We need to break Fg up into components that point down parallel to the slope (F//) and perpendicular to the slope (F).

Figure 2

If you are wondering how I figured out that the 30°angle is at the top of the red triangle, take a look at this...

Figure 3

Since we know Fg and an angle on the triangle, we can use basic trig to calculate the other two sides.

Only round off a number if it's a final answer, like if this was part (a) of a bunch of questions together. Otherwise, leave it as is so you aren't looking at the rounded off answer of a rounded off answer of a etc etc etc...

Determine F//...

sinΘ = opp / hyp
sinΘ = F// / Fg
F// = sinΘ Fg
= sin30° (147.15 N)
F// = 73.575 N
(Again, we'll keep the number unrounded. This is the force pulling it down along the slope.)

Determine F...

cosΘ = adj / hyp
cosΘ = F / F
F = cosΘ Fg
= cos30° (147.15 N)
F = 127.436 N
(Guess what, don't round off yet... or anything else right up to the end. Oh, and by the way, look back at Figure 2... F is equal to the normal force, FN!)

Determine the force due to friction using the value you just got for normal force.

Ff = μFN
= 0.15 (127.436)
Ff = 19.115 N

Now you know the force that is taking it down the slope, and the friction that is slowing it down. Determine the net force FNET...

I've made Ff negative because it is working against the F//. One of them must be negative if the other is positive.

FNET = F// + Ff
= 73.575 + -19.115
FNET = 54.460 N

Now, finally we can determine the acceleration of the box...

FNET = ma
a = FNET / m
= 54.460 / 15
a = 3.6 m/s2

Our final answer is 3.6 m/s2 , and yes, we round it off.

Notice that in each step I had you sketch or determine something.

Re-read through this example a few times. It's long and confusing at some parts, but try to look at each individual tiny calculation. Taken in little bits each part isn't as hard.