Figure shows a block (mass ) on a smooth horizontal surface, connected by a thin cord that passes over a pulley to a second block which hangs vertically. (a) Draw a free-body diagram for each block, showing the force of gravity on each, the force (tension) exerted by the cord, and any normal force. (b) Apply Newton’s second law to find formulas for the acceleration of the system and for the tension in the cord. Ignore friction and the masses of the pulley and cord.
m1 up top
m2 falling
tension T
distance m2 moves DOWN is x is also distance m1 moves horizontal (string does not stretch)
now for m1
force down on table = m1 g
force up from table = m1 g
no vertical acceleration
only force horizontal is T
so horizontal acceleration = a1 = T/m1
now for m2
no horizontal force or acceleration
force down = m2 g
force up = T
total force down = m2 g - T
acceleration a2 = (m2 g -T)/ m2
now if the string does not stretch then a2 = a1
T/m1 = (m2 g - T)/m2
T (m2/m1) = m2 g - T
T (1 + m2/g) = m2 g
now if by any wild chance (to check answer) m2 = m1 = m
2 T = m g
T = mg/2 LOL, yes accelerates half as fast as one would alone :)
(a) Sure, here are the free-body diagrams for each block:
For the block on the smooth surface:
- Force of gravity points downwards.
- Normal force (since it's on a smooth surface) is equal in magnitude and opposite in direction to the force of gravity.
- There is no friction, so we don't need to include it.
- The tension in the cord is directed towards the pulley.
For the hanging block:
- Force of gravity points downwards.
- There is no normal force, as it is not on a surface.
- The tension in the cord is directed away from the pulley.
(b) Applying Newton's second law, let's call the mass of the block on the smooth surface "M" and the mass of the hanging block "m". The formulas for acceleration and tension are as follows:
Acceleration of the system: a = (m * g) / (M + m)
Tension in the cord: T = M * g - m * a
Note: Just a friendly reminder that these formulas are valid only in ideal conditions, ignoring friction and the masses of the pulley and cord.
(a) Free-body diagrams:
For the block on the smooth horizontal surface:
- Force of gravity (weight) down;
- Normal force (if applicable);
- Tension in the cord.
🔲-----------------
| M1
| ↓
|
| Tension
| ↑
|
| ⬆️ Normal force (if applicable)
|
🔲---------------
For the block hanging vertically:
- Force of gravity (weight) down;
- Tension in the cord.
🔲 ⬇️
|
| M2
| ↓
▬▬▬▬▬Tension
| ↑
|
| Force of gravity (weight)
|
🔲 ⬆️
(b) Newton's second law for the block on the horizontal surface:
Net force = mass × acceleration
Since there is no acceleration in the horizontal direction:
Net force = 0 = Tension - weight
Tension = weight
For the block hanging vertically:
Net force = mass × acceleration
Since the only external force acting on the block is the tension in the cord:
Net force = Tension - weight
Tension = weight + (mass × acceleration)
Since the two blocks have the same acceleration (since they are connected by the same cord), we can equate the tension equations:
weight = weight + (mass × acceleration)
Simplifying, we get:
(mass × acceleration) = weight
acceleration = weight / mass
The tension in the cord can be found by substituting the acceleration back into either of the tension equations.
(a) To draw a free-body diagram for each block, we need to consider the forces acting on each block:
1. Block 1 (on the horizontal surface):
- Force of gravity (mg): This is the weight of the block acting vertically downward.
- Tension force in the cord (T): The cord pulls the block to the right.
2. Block 2 (hanging vertically):
- Force of gravity (Mg): This is the weight of the block acting vertically downward.
- Tension force in the cord (T): The cord pulls the block upwards.
Additionally, there are no other forces acting on the blocks since friction and masses of the pulley and cord are ignored.
(b) Applying Newton's second law to each block:
1. Block 1:
The net force acting on block 1 is given by:
Net force = T - mg
Since the block is on a smooth horizontal surface, there is no acceleration in the horizontal direction. Therefore, the net force is balanced by the normal force:
T - mg = 0
2. Block 2:
The net force acting on block 2 is given by:
Net force = T - Mg
Due to the acceleration, the net force is equal to Ma, where "a" is the acceleration of the system. Since block 2 is hanging vertically:
T - Mg = Ma
Now, we have two equations:
1) T - mg = 0
2) T - Mg = Ma
Solving these two equations will yield the formulas for acceleration (a) and tension (T) in terms of the given masses (m and M) and the acceleration of the system.