Friction, Inclined Plane, and Pulley HW question?

These problems are modified versions of the Atwood machine.

They are best approached looking at free body diagrams (FBDs) of each component and then applying F=m*a

When the pulley is massless and frictionless, there are some simplifications to the problem.

First, the tension in the cord is the same everywhere

(there is a special case where that is not true. I won't go into it here since it may confuse. If you want to know more about the case of "slack cord", send me an email).

Second, the acceleration of the masses are equal.

Okay, I like to start with a FBD of the hanging mass. Since it is much more massive than the one on the slope, I will make the assumption that the mass will descend when released from rest if the tension is sufficient to break static friction on the slope. So positive will be in that direction of motion

50*9.81-T=50*a

T=50(9.81-a)

Now let's look at the mass on the slope.

The forces parallel to the slope are

T, the tension in the cord

10*9.81*sin(30), the force of gravity parallel to the slope

10*9.81*cos(30)*µ, this is the force of friction

now let's set up F=m*a and test to see if the system will accelerate.

T-10*9.81*sin(30)-

10*9.81*cos(30)*µ=10*a

simplify

T=10*a+49.05+84.96*µ

from above

50*(9.81-a)=10*a+49.05+84.96*µ

solve for a

a=(441.45-84.96*µ)/60

Ok,

let's test to see if we get a>0 when µ=0.6

a=6.51 so it will move

Once moving, µ=0.5, so the actual

a=6.65

Just for fun, let's find T from the FBD of the hanging mass

158 N

Check by plugging T and a into the FBD equation for

the mass on the incline

You will get the two sides equal if all of the math and algebra are done right. I get 66.5=66.5

j

it is through practise you will be able to solve such problems.

First you have to find out whether the mass on the inclined plane will move or not. It will move only if the tension in the string due to hanging mass exceeds the limiting static frictional force of the mass on the plane.

Assuming that all the masses are stationary, the tension in the string is equal to weight of the handing mass = 490N.

Maximum static frictional force for the mass on the plane is

.6*Mg*cos 30= 50.9N.

As 490N>50.9N, we can conclude that the masses will move. Due to motion the frictional force on the mass on the slope will be less and it will be .5*Mg*cos 30= 42.48.

Now you have to draw free body diagrams for both the masses and eliminate the uknown Tension in the string and find the acceleration of the hanging mass. As it is your homework i will let you do that :)