Dynamics Help!

[DavidSheena]

A mass of 10kg sits on a smooth horizontal plane and is connected by a light inextensible string, which passes over a smooth pulley, to a mass of 12kg. This 12kg mass sits on a rough plane with coefficient of friction (sqrt(3)/10) and which is inclined at an angle of 30 degrees to the horizontal. The tension in the string is T newtons. A horizontal pulling force of P newtons acts on the 10kg mass.

The 12kg mass is at rest for values of P such that Q<P<R.

Find Q and R

Please help!

[BubbleWrapMan]

Without giving you the entire answer, just note that when the 12 kg mass is on the point of sliding down the plane (when P = Q), friction is acting up the plane, and when it is on the point of sliding up the plane (when P = R), friction is acting down the plane. This gives you two different equations of motion which you can solve for Q and R.

[DavidSheena]

I tried it, but I still have no idea. ><

[BubbleWrapMan]

Okay, well

If you take the motion up the plane as positive you get:

[DavidSheena]

Thank you so much! Why does the Tension in the string cancel out like that?

Also, following the same question:

While the 12kg mass is being hauled up the inclined plane, the string is suddenly disconnected. Simultaneously, the mass is given a one-off push causing it to have an initial velocity of 1m/s down the inclined plane.
What sort of conditions does the string being 'suddenly disconnected' result in? I'm confused with the types of questions where a string suddenly snaps and whatnot.

a) How long after the rope is disconnected does the 12kg mass take to travel the 8m to the end of the inclined plane?
b) How long after the rope is disconnected would it take for the 12kg mass to travel the 8m to the end of the plane if it had not received the one-off push?

Sorry about the barrage of questions.

[BubbleWrapMan]

The tension cancels out because it just connects the system without doing much else -- as long as the string is taut, the two objects connected will experience the same acceleration so in this sort of case it doesn't matter what the tension is, but of course there are cases where it does matter. Note that the same tension runs through the entire string (if it's taut), or else the string would accelerate relative to the rest of the system.

"Suddenly disconnected" just means T is no longer acting on the 12 kg mass, so basically you just need draw a new diagram and see what forces are left to act on it. It's probably a good idea this time to take motion down the plane as positive for ease of calculation. In retrospect I probably just would have taken downward as positive to begin with. =P

For a), use the equation s = ut + (1/2)at^2 (you have u = 1 m/s, you can work out a from calculating the net force, and you have s = 8 m)

For b) do the same thing except with u = 0 m/s

Note that if it works out that the (maximum) frictional force (µN) is greater than the resultant weight force (12gsin(30)) then the object will not move, even though you'd calculate an acceleration up the plane (a negative acceleration). This is because friction will only be as big as it needs to be (or else objects would start moving on their own), until it reaches its maximum value of µN, which is something you need to be conscious of in your calculations. If you assume friction is ALWAYS µN you might calculate an acceleration up the plane, but nothing is actually causing it to move up. I hope that makes sense. That said, you'll probably get a positive acceleration, but it was worth mentioning.

[DavidSheena]

Everything makes a lot more sense now. Thanks so much for taking the time out to help me, really appreciate it

[BubbleWrapMan]

No worries, if any of it's still unclear in the slightest don't hesitate to ask.

[DavidSheena]

If an object is on the verge of moving up a plane, inclined at whatever angle, will the force of friction always be opposite (down the plane) regardless of the weight of the object?

[BubbleWrapMan]

Friction is always opposite to the direction of motion (or the direction of a force that may cause motion, i.e. on the verge of sliding as you said). That's essentially the definition of friction -- resistance to motion.

[DavidSheena]

Cheers man! And if there's just a large mass (say 70kg) on a rough inclined plane. Is there a tendency for the mass to slide down the plane? In other words, in addition to the normal force and the weight, would there be a force parallel down the plane and a friction force up the plane?

Sorry about all the questions ><

[BubbleWrapMan]

There wouldn't be a "tendency" as you say. It would just require more force to cause it to start moving upward and cause a downward friction.

For an example, replace the 12 kg mass with a 70 kg mass. For the first part of the question, you will get these equations:





There is still a value of R so that the 70 kg mass will slide up the plane, but it is noticeably larger than the force needed for the 12 kg mass. It doesn't really matter how heavy the object is, there will always be a force that can move it up the plane (though realistically the string would probably break). If you rearrange the above equations and just use a variable m instead of 70, you will get:





In other words the forces are directly proportional to the mass of the object.