In the drawing, the weight of the block on the table is 430 N and that of the hanging block is 165 N. Ignore all frictional effects, and assuming the pulley to be massless.
(a) Find the acceleration of the two blocks.
(b) Find the tension in the cord.
To solve this problem, we can use 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.
(a) Find the acceleration of the two blocks:
1. Start by analyzing the forces acting on each block separately.
- For the block on the table:
- The weight of the block (acting downwards) is 430 N.
- The tension in the cord (acting upwards) is T.
- For the hanging block:
- The weight of the block (acting downwards) is 165 N.
- The tension in the cord (acting upwards) is T.
2. Next, write the net force equations for each block:
- For the block on the table:
Net Force = Ma
(T - 430 N) = M1 * a1
- For the hanging block:
Net Force = Ma
(165 N - T) = M2 * a2
3. Since the two blocks are connected by the same cord, they will have the same acceleration:
a1 = a2 = a (let's call it "a" for simplicity)
4. Now, we can solve the system of equations:
(T - 430 N) = M1 * a
(165 N - T) = M2 * a
We also know that the masses are related by the equation:
M1 = M2
5. Let's substitute M1 with M2 in the first equation:
(T - 430 N) = M2 * a
6. Now, add the two equations together to eliminate "T":
(T - 430 N) + (165 N - T) = M2 * a + M2 * a
-265 N = 2M2 * a
7. Simplify:
-265 N = 2M2 * a
8. Divide both sides of the equation by 2a:
-265 N / (2a) = M2
9. Now we have an equation for M2 in terms of "a". We can substitute this back into one of the previous equations to solve for "T".
(b) Find the tension in the cord:
1. Substitute the value of M2 from step 9 into one of the previous equations:
(T - 430 N) = M2 * a
2. Substitute the value of M2:
(T - 430 N) = (-265 N / (2a)) * a
3. Simplify:
T - 430 N = -265 N / 2
T = -265 N / 2 + 430 N
4. Calculate the tension in the cord:
T = -265 N / 2 + 430 N
T = -265 N / 2 + 860 N
So, the tension in the cord is 295 N.
To find the acceleration of the two blocks and the tension in the cord, we can use Newton's second law of motion.
(a) Acceleration of the two blocks:
First, let's analyze the forces acting on each block separately:
1. For the block on the table:
The weight of the block acts downwards with a force of 430 N. Since there is no friction, the only force acting in the horizontal direction is the tension in the cord, which we'll call T1.
2. For the hanging block:
The weight of the hanging block acts downwards with a force of 165 N. The tension in the cord, T2, acts upwards.
Now, we can apply Newton's second law to each block:
1. For the block on the table:
The net force in the horizontal direction is given by:
Net force = T1 - weight of the block = T1 - 430 N
Using Newton's second law:
Net force = mass * acceleration
Since the mass is not given, we can express it in terms of weight:
mass = weight / gravitational acceleration = 430 N / 9.8 m/s^2
Now we have:
T1 - 430 N = (430 N / 9.8 m/s^2) * acceleration
2. For the hanging block:
The net force in the vertical direction is given by:
Net force = T2 - weight of the block = T2 - 165 N
Using Newton's second law:
Net force = mass * acceleration
Again, we can express the mass in terms of weight:
mass = weight / gravitational acceleration = 165 N / 9.8 m/s^2
Now we have:
T2 - 165 N = (165 N / 9.8 m/s^2) * acceleration
Since the two blocks are connected by the same cord and the cord is assumed to be massless, the acceleration of both blocks will be the same. Therefore, we can set the two equations equal to each other:
T1 - 430 N = (430 N / 9.8 m/s^2) * acceleration = T2 - 165 N
Simplifying the equation, we get:
T1 - T2 = (430 N / 9.8 m/s^2) * acceleration + 165 N - 430 N
(b) Tension in the cord:
To find the tension in the cord, we can solve for T1 or T2 using the equation derived above. Let's solve for T1:
T1 = (430 N / 9.8 m/s^2) * acceleration + 165 N - 430 N + T2
Once we have calculated the acceleration from part (a), we can substitute this value back into the equation to find T1 or T2, which represents the tension in the cord.