Understanding Newton's Second Law in a System of Two Connected Blocks

[sweet-buds]

urgent:Newton2nd law

Homework Statement

i got this problem in Haliday. There are 2 blocks connected by a rope. onr of the blocks is a sliding one and the other is a hanging one and the rope passes over a pulley and the pulley is connected at the edge of the table/surface. The sliding block moves towards the right and the hanging block down. Haliday has applied the Newtons second law to the two blocks.

The sliding block has mass M and hanging block a mass of m. haliday has said the the equation for hanging block is T-mg =-ma. i understand the fact that the hanging block is moving down in -ve y directio, that's why a istaken -ve here but doesn't that mean that mg>T for hanging block to move down. I mean shouldn't the equation be mg-T=-ma?

Homework Equations

The Attempt at a Solution

 

[berkeman]

I think that the positive direction for that equation is up, so T points up (the rope pulls up on the block), and the force due to gravity points down. So Haliday's equation looks correct to me.

 

[sweet-buds]

if T points up and mg down, and if the block moves down then mg>T
so can the equation be mg-T= m*(-a) as acceleartion is -ve as it is a vector.

 

[sweet-buds]

anyone?PLs reply

 

[Gear300]

mg is the positive one because it is the direction the block that is hanging is going. Usually, to be safe, when claiming positive and negatives, positive is usually with the movement. Consider it like this, -(a) is deceleration; a system like this has a constant acceleration, so if you were to consider up as positive, down as negative, right as positive, and left as negative (Cartesian system), then you would end up with a -(a) for the block going down and a +(a) for the block going to the right. So, best to thing to take note of is that positive is in direction of motion.

 

[sweet-buds]

what do u mean by your last sentence "positive is in direction of motion" is it the acceleration. suppose ----->[]<--------- the forces act on the box(mass m) as shown
3N 5N

what would be the equation? would it be :-5N+ 3N=-2N(NET FORCE)
So, -2N= m*a or m*(-a).According to you
r note I would say, -2=m*a as acceleration is positive in direction of motion.

lets say the forces act ------>5N[] 3N<------
In this case, the equation according to me should be 5-3=m*a=>2=m*a.

in both cases, unknown acceleration should be positive in the direction of motion as u said although acceleration would come out to be negative in the first case. Am I correct?

 

[Gear300]

The thing is to not to place everything on a cartesian plot, in which right is positive, left is negative, etc... If an object is moving in a direction, that is usually considered positive. So if 5N and 3N were two opposing forces, then it would be 5N - 3N = 2N not -5N +3N because the Net Force shows that 5N > 3N, so the force 5N would be moving the box, making 5N the positive value. Thats how the hanging object in the question should be attended to, in which the net force is pulling that mass downward (only gravity is acting on it, no air resistance, so it is going downward). If it is moving downward, then it is accelerating in that direction and mg is positive because it is the force that is moving the box.
A negative acceleration is referred to as deceleration or the acceleration is in the opposite direction of the motion. In this case, the box that is hanging is accelerating downward.

 

[sweet-buds]

sorry, I still didnt get you. could explain it more elaborately.

 

[Gear300]

Example: If you take a box and have a force of 5N pushing to the right and 3N to the left, then it moves right not left. In this case, an object tends to move in the direction the net force is pointing towards, meaning that since 5N is greater than 3N, it moves in that direction. Because positive is usually taken in the direction of movement, then the object 5N is positive and 3N is negative. That is how the hanging rock should be looked at. It is hanging down and nothing is moving it up but tension. It is already given that the hanging block is moving down, so that means the force of gravity is greater than the force of tension, meaning mg should be taken as positive. The block on the table is moving right due to tension (remember that tension on one side of the rope is the same on the other if the rope is given mass = 0, so the tension pulling it should be the same as the tension pulling on the hanging block). The only force opposing this block is friction (acting to the left), so if the block is moving right then the acceleration to the right is positive and the force of friction (opposite of motion) is negative.
Try to look at each block independently and understand the forces influencing each box separately as well (free-body diagrams).

 

[sweet-buds]

so u mean if the forces 5N acted towards left and 3N towards right, it would move left. In this case waht would be Newtons 2nd law equation?
I would say since 5N>3N, So left direction would be positive and so 2N=m*a or 2N= m*(-a) as the body tend to move left. I would go with the former equation.

 

[Gear300]

Yes.
Newton's Second Law of Motion: "The acceleration of an object is directly proportional to the net force acting on it, and is inversely proportional to its mass. The direction of the acceleration is in the direction of the net force acting on the object" This is basically a = (Fn)/m or Fn = ma. Its basically saying that if an object is in its inertial state (state of rest or constant velocity) then to apply motion to the object or change its direction of motion a force is needed, in which it the net forces will add up to a direction, in which it accelerates. If an object were at rest, then its velocity is 0; to move it, it needs a force, in which it accelerates (the velocity has to move from 0 to another value). Thats basically the law.