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In the system below there is friction between mass m1 and mass m2. There is also friction between mass m2 and the floor. The coefficient of friction (static is considered the same as kinetic) of both surfaces is the same, namely μ. The pulley is massless and the rope cannot stretch.
a) Draw a diagram of the forces on object 1 and object 2?
b) Write the equation of motion of object 1 and object 2?
c) How much external force F is needed so that the system can move at a constant speed?

#IPHO #PhysicsOlympiad

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00:00Hi friends, welcome to those of you who have just found this channel.
00:08This channel is one of the channels that provides educational materials in the form of animations.
00:17This is the question that we will discuss in this series.
00:21The question is quite long.
00:23You can read this question for 10 seconds.
00:38Do you understand this question correctly?
00:42Let's watch the following short animation.
00:46There are two stacked blocks connected by a rope.
00:50The rope connecting the two blocks is passed to a pulley.
00:54Thus, when the rope moves, the movement of the rope can rotate the pulley.
01:04Then the block at the bottom is pulled to the right by an external force.
01:09Of course the block pulls the rope behind it.
01:14The pull of the rope then pulls the block above the block.
01:20So the two blocks move in opposite directions.
01:23If the lower block shifts to the right, the upper block will move to the left.
01:31Now we are asked to draw the force lines acting on this system.
01:37Of course the equation of motion of the block.
01:41And the last is the magnitude of the external force needed for the system to move at a constant
01:46speed.
01:50Now we understand.
01:54Let's start answering the first part of this question.
02:00Before going further, we will identify some important quantities.
02:06Let's say the mass of the upper block is m1.
02:09The mass of the lower block is m2.
02:14The surface of the block and the floor are not slippery.
02:17There is a coefficient of static friction.
02:23The magnitude of the coefficient of static friction between the surface of the block
02:27and the floor and between the surface of the block and the block is the same, namely mu.
02:34For the pulley, there is no information in the question.
02:38We can ignore the movement of the pulley.
02:44An object moves or does not move because it is influenced by the force acting on the object.
02:51For the upper block, this block gets a gravitational force of m1g downwards.
02:58This block does not penetrate the surface of the block below it.
03:02Of course, block 1 gets a normal force and 1 upwards.
03:08This block is connected to a rope.
03:10There is a rope tension force.
03:12The direction is in line with the rope.
03:17Because both surfaces of the block are rough, the upper block will get a static friction
03:21force to the right, Fk1.
03:25It seems that this is the only force diagram for block 1.
03:31For the lower block, we do not know the magnitude of the external force.
03:35The force is F.
03:40Block 2 is also connected to the rope.
03:42There is also a rope tension force.
03:47For the surface of the block and the surface of the floor, there is a friction force Fk2
03:52to the left.
03:55The magnitude of the gravitational force acting on the lower block is m2g.
04:03Is this the only force acting on block 2?
04:06Let's look again more carefully.
04:10The force is an external action.
04:14Where there is action, there is reaction.
04:17This is true, isn't it?
04:21If at the top a force Fk1 works to the right, of course a force of the same magnitude and
04:26opposite direction will also work on block 2.
04:31This means that block 2 also works force Fk1, but the direction is opposite to the
04:36direction of Fk1 on block 1.
04:40You can understand this, if you have time to watch the motion dynamics course on this
04:45channel.
04:47Furthermore, the upper block presses on the lower block.
04:52Thus, the molecules in the lower block push the upper block with a force known as the
04:57normal force, N1.
05:01This force N1 arises due to the pressure from the upper block.
05:07This means that the lower block also has a force N1, but its direction is opposite to
05:12the force N1 acting on block M1.
05:18One more thing, the lower block presses on the floor surface.
05:22Of course the floor surface will provide a reaction force that is directed upwards.
05:27This force is none other than the normal force N2, which is directed upwards.
05:34So on the lower block there are two normal forces working at once.
05:41This is the force diagram in this system.
05:47Now we will answer the second part.
05:51Newton's laws work for every object.
05:54Different objects apply different Newton's laws.
06:00For the first block, we will write the equation in two directions, the vertical direction
06:04and the horizontal direction.
06:08For the vertical direction, the block does not move up or down.
06:14The resultant force in this direction is zero, and 1-M1g is equal to zero.
06:23For the horizontal direction, the block moves to the left.
06:26So the direction to the left is positive.
06:30In motion analysis, the direction that is in the same direction as the direction of
06:34motion has a positive sign.
06:37T-fk1 is equal to M1a.
06:41Since the question has mentioned that this system is moving at a constant speed, the
06:46acceleration value is zero.
06:51For block 2, we will also analyze the resultant force in two directions.
06:57For the vertical direction, there are three units of force.
07:00N2-N1-N2g is equal to zero.
07:07For the horizontal direction, block 2 moves to the right.
07:16F-fk1-fk2-t is equal to zero.
07:21The resultant force is zero, because this system is moving at a constant speed.
07:29These are the four equations of motion for objects M1 and M2.
07:34Knowing these four equations, we can calculate the magnitude of the external force.
07:42We start from the equation that looks the simplest.
07:47Let's write, from equation 1, N1 is equal to M1g.
07:54Next is this equation.
07:59T is equal to fk1.
08:04The friction force is mu multiplied by the normal force.
08:10T is equal to mu M1g.
08:15Now is this equation.
08:19N2 is equal to N1 plus M2g.
08:25We already know the value of N1.
08:30N2 is equal to M1 plus M2, multiplied by g.
08:37Next is this equation.
08:42F is equal to fk1 plus fk2 plus t.
08:48All of these values are also known.
08:53F is equal to mu times 3M1 plus M2, g.
09:00This is the external force required to pull the block so that the system moves at constant velocity.
09:09Hope this helps.
09:11Don't forget to watch the next video.
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