9. Two blocks connected by a cord passing over a small frictionless pulley rest on frictionless planes as shown in the figure below.
(a) Which way will the system move?
(b) What is the acceleration of the block?
(c) What is the tension in the cord?
Physics
9. Two blocks connected by a cord passing over a small frictionless pulley rest on frictionless planes as shown in the figure below.
(a) Which way will the system move?
(b) What is the acceleration of the block?
(c) What is the tension in the cord?
To determine the direction in which the system will move, we need to compare the masses of the two blocks. Let's call the mass of the left block "m1" and the mass of the right block "m2".
(a) If m1 > m2, the system will move towards the right.
If m1 < m2, the system will move towards the left.
If m1 = m2, the system will not move because the forces on both sides are balanced.
(b) To find the acceleration, let's denote the tension in the cord as "T". The net force acting on each block can be calculated using Newton's second law:
For the left block:
F_net = m1 * a1, where a1 is the acceleration of the left block
For the right block:
F_net = m2 * a2, where a2 is the acceleration of the right block
Since the tension in the cord is the same on both sides, we have:
T = m1 * a1 = m2 * a2
We also know that the acceleration of the two blocks are equal in magnitude but opposite in direction, so we can write:
a1 = -a2
Substituting this into the equation above gives:
T = m1 * (-a2) = -m1 * a2
Now, let's combine the two net force equations to eliminate the acceleration:
m1 * a1 = -m1 * a2
Rearranging this equation gives:
a1 / a2 = -m1 / m2
Since the acceleration ratio is equal to the inverse of the mass ratio, we conclude that the acceleration of the system is given by:
a = (m2 - m1) / (m2 + m1)
(c) Now let's calculate the tension in the cord. For this, we use Newton's second law on one of the blocks (e.g., the left block):
F_net = m1 * a1
The net force acting on the left block has two components: the force due to the tension in the cord pulling it upwards, and the force due to its weight pulling it downwards. Thus, we have:
F_net = T - m1 * g, where g is the acceleration due to gravity
Substituting the expression for the tension obtained earlier, we get:
T = m1 * a1 + m1 * g
Using the value of a1 obtained from part (b), we can calculate the tension.
To answer the questions, let's break down the problem step by step:
(a) Which way will the system move?
To determine the direction of motion, we need to consider the net force acting on the system. In this case, there are two forces to consider: the force of gravity and the tension in the cord.
Since the block on the left has a greater mass than the block on the right, the force of gravity on the left block is greater. This means that the net force will be directed towards the right. Therefore, the system will move to the right.
(b) What is the acceleration of the block?
To find the acceleration, we need to apply Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force is the difference between the force of gravity on the left block and the tension in the cord. Let's denote the mass of the left block as m1, the mass of the right block as m2, and the acceleration as a.
Net force = m1 * g - T
The force of gravity on the left block is m1 * g, where g is the acceleration due to gravity. T represents the tension in the cord. Since the system is moving to the right, the net force is positive.
Using Newton's second law:
m1 * g - T = m1 * a
(c) What is the tension in the cord?
To find the tension in the cord, we need to consider the motion of the two blocks as a whole. Since the blocks are connected by the cord, they have the same acceleration.
Applying Newton's second law to the right block:
T - m2 * g = m2 * a
Since the acceleration of both blocks is the same, we can equate the two equations:
m1 * g - T = T - m2 * g
Rearranging the equation, we find:
T = (m1 + m2) * g / 2
Thus, the tension in the cord can be calculated using the sum of the masses of the two blocks and the acceleration due to gravity, divided by 2.