The two blocks in Fig. P4.48 are connected by a heavy uniform rope with a mass of 4.00 kg. An up

The two blocks in Fig. P4.48 are connected by a heavy...

Key Concepts

System Dynamics

System dynamics involves analyzing all the forces and masses in a composite system to understand its overall motion. By summing the forces acting on the interconnected parts and applying Newton's laws, one can determine the system's acceleration and the internal forces such as the varying tension in different sections of a massive rope.

Tension Distribution in Massive Ropes

Tension distribution in a rope that has mass is non-uniform because each segment of the rope must support the weight of the rope below it, in addition to any external forces. Understanding how tension varies within such a rope is essential for solving problems related to the forces acting at different points along the rope.

Free Body Diagrams

Free body diagrams are schematic representations of an object isolated from its surroundings, showing all the external forces acting upon it. They are crucial for visually breaking down complex systems into simpler parts, making it easier to apply physical laws such as Newton's Second Law for solving problems in mechanics.

Newton's Second Law

Newton's Second Law of Motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma), is a fundamental principle used to relate force, mass, and acceleration. This law provides the mathematical framework for analyzing the dynamics of the entire system and its individual components.

Non-negligible Mass Effects in Ropes

When a rope has a significant mass, its weight affects the dynamics of the system. Unlike an idealized massless rope where tension is uniform throughout, a rope with mass exhibits a variation in tension along its length. This concept is important for accurately analyzing forces in systems involving real, massive ropes.

Transcript

00:01 Okay, so we have two masses on a string problem, but the gimmick of this problem is that the string itself has mass, so that will complicate things a little bit.

00:10 So this is a vertical type of problem.

00:12 So we have a mass up here, and then i'm going to draw the rope like that, and then a second mass down here.

00:20 So we are drawing three body diagrams for all of these bodies, and i find it easiest to definitely not start in the middle, but either start at the top of the bottom and start from the bottom.

00:30 Because gravity acts downward.

00:32 So i'm going to call this blocks or item c in this problem.

00:36 It's going to have a gravitational force directed downward.

00:40 And then i'm going to call this a second tension force t2 that's acting upward because it's been pulled along by a rope.

00:48 And then working our way upward, we know that there must be a newton's third law reaction pair.

00:54 So that's tension two also pointing downward.

00:57 Force of gravity is also involved.

01:00 Mbg and then a potentially different tension force, t1, acting upward.

01:07 And similarly, in block a, we have the tension force also acting downward, gravity, acting downward, and then an applied force acting upward.

01:19 So let's call that f.

01:22 Okay.

01:23 Additionally, the problem wants us to explicitly state for all the forces, what by is exerting that force.

01:33 So i'd say the easiest one to do first is all the gravities and earth is exerting the gravitational force on all three of those.

01:48 On the tension forces it's just coming from the adjacent or neighboring body.

01:52 So for t2 on item 3, that would be b and vice versa and then for t1 it's coming from a and then up here it's coming from b and then honestly based on the grammar of the problem they don't tell us what's causing the external applied for us so we can put yourself maybe you're holding it by a string but i'm just going to put a question mark there because it's ambiguous as to what it is um in part b we want to find the acceleration of the system is so we'll go ahead and do that by turning our three for your body diagrams into three equations of motion using newton's second law so we have f is equal to m a for block one so the sum of the forces is f minus t1 minus m -a -g and do similar things for blocks b and c.

02:46 And note that the accelerations are all the same because they're connected by a string, so they all need to be moving and therefore accelerating together.

02:59 T -2 minus m -c -g.

03:03 Okay, and i think the easiest way to solve for acceleration is just add these three equations together.

03:10 So on the left -hand side, if i collect terms, i get the total mass.

03:15 Multiplied by their joint acceleration.

03:18 And then we see that the tensions all cancel out if i add them on this side.

03:21 So i'm just left with the applied force, minus, again, the sum of the masses, and multiplied by gravity.

03:32 Okay, so now i can solve for acceleration by dividing both sides by total mass minus g.

03:47 And then we plug in numbers...

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