Interpret results of suspended mass / tension problem please

[Tyrannosaurus_]

Homework Statement

Two masses: m1 = 45 kg, m2 = 12kg.
m1 (45kg) sits at rest on a table top with static friction co. 0.45, kinetic friction co. 0.35.
m2 is suspended in the air. m2 is attached to m1 by a massless wire hung over a frictionless pulley

Homework Equations

FBDs, net force equations

The Attempt at a Solution

m1 is at rest, so I'll use the μs = 0.45.
I've attached an image of my solution.

My question is how do I interpret the negative acceleration value? I know it's a vector, but it's not as though the simple mass that is sitting on the table can accelerate, and even pull the 12kg mass even further higher into the air. So, does this mean that the system is at rest and there would be no acceleration because Fg is not greater than the Fk?

Many thanks!

 

[haruspex]

m1 is at rest, so I'll use the μs = 0.45.

That only makes sense as far as determining whether the system will remain at rest. If not, you need to switch to kinetic friction to discover the acceleration.
Your working is hard to read. You should only post images for diagrams and textbook extracts. Please take the trouble to type other working in directly. Not only is it easier to read, it's easier to identify sections of working when posting replies.

how do I interpret the negative acceleration value?

That probably arises from assuming the frictional force equals μ_sN. μ_sN is merely the maximum possible frictional force. In equilibrium, it will usually be less than that.

 

[Nathanael]

It's too difficult to follow your work, but I think I can deduce the error you made from your answer. You treated m₁gµ_s as if it were the actual force of friction acting. This is not correct; m₁gµ_s represents the maximum force that static friction is capable of exerting. The actual force of static friction will only be as much it needs to be to keep the block at rest (unless of course m₁gµ_s is not large enough, in which case the system will accelerate and only µ_k will be relevant).

 

[Tyrannosaurus_]

Thanks for the help! Much appreciated!

 

[HalfLostAndSearching]

I would interpret the results of this suspended mass/tension problem by first looking at the forces acting on the system. From the given information, it appears that the only forces acting on the system are the weight of the two masses (m1 and m2) and the tension in the massless wire.

Using the equations for net force, we can see that the net force on m1 is equal to the weight of m1 minus the frictional force. Since the table is assumed to be stationary, there is no net force on m1 and it remains at rest.

On the other hand, the net force on m2 is equal to the weight of m2 minus the tension in the wire. Since m2 is suspended in the air, there is no contact with any surface and therefore no frictional force. This means that the net force on m2 is equal to its weight, which causes it to accelerate downwards.

The negative acceleration value simply indicates that the direction of motion is downwards, which is consistent with the direction of the net force on m2. This does not mean that the system is at rest, as the acceleration of m2 is causing it to move downwards.

In terms of interpreting the negative acceleration value in relation to the system as a whole, it means that m2 is pulling m1 downwards with a force equal to its weight. This is because the massless wire, being hung over a frictionless pulley, transmits the tension force from m2 to m1 without any loss. Therefore, the system as a whole is in motion due to the acceleration of m2, with m1 being pulled downwards at the same rate.