A 90.0 kg block and a 40.0 kg block are connected by a rope that passes through two (frictionless, massless) pulleys as shown. When released, what is the acceleration of the larger block? What is the tension in the rope?
To find the acceleration of the larger block, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the larger block has a mass of 90.0 kg. Let's call the acceleration of the larger block 'a'.
We need to note that both blocks are connected by a rope that passes through two (frictionless, massless) pulleys. This means that the tension in the rope is the same throughout the system.
Now, let's analyze the forces acting on the larger block. The weight of the larger block (due to gravity) acts downwards with a force given by F = m*g, where m is the mass of the larger block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The tension in the rope will act upwards, opposing the weight of the block.
Since the two blocks are connected by the same rope, the tension will also act on the smaller block.
The net force on the larger block is given by the difference between the downward weight force and the upward tension force. So we can write:
F_net = m*a = m*g - T,
where T is the tension force.
Similarly, for the smaller block, we can write:
F_net = m*a = T,
where m is the mass of the smaller block.
Since the tension force is the same in both cases, we can set the two equations equal to each other:
m_1 * a = m_1 * g - T,
m_2 * a = T,
where m_1 = 90.0 kg and m_2 = 40.0 kg.
Simplifying the equation, we get:
90.0 * a = 90.0 * 9.8 - T,
40.0 * a = T.
Now we have a system of equations. Solving these equations simultaneously will give us the acceleration of the larger block and the tension in the rope.
To do this, let's substitute the second equation into the first equation:
90.0 * a = 90.0 * 9.8 - 40.0 * a.
Simplifying further:
130.0 * a = 90.0 * 9.8.
Dividing both sides by 130.0:
a = 90.0 * 9.8 / 130.0,
a ≈ 6.77 m/s^2.
To find the tension force, substitute the value of 'a' into the second equation:
40.0 * a = T.
T = 40.0 * 6.77,
T ≈ 271 N.
Hence, the acceleration of the larger block is approximately 6.77 m/s^2, and the tension in the rope is approximately 271 N.
To find the acceleration of the larger block and the tension in the rope, we can use Newton's second law of motion and apply it to both blocks.
Step 1: Identify the forces acting on the blocks:
- The weight of each block acting downwards (mg)
- The tension in the rope acting upwards for both blocks.
Step 2: Set up the equations of motion for each block:
For the 90.0 kg block:
Sum of forces = mass x acceleration
Tension in the rope - weight of the block = mass x acceleration
For the 40.0 kg block:
Sum of Forces = mass x acceleration
Tension in the rope - weight of the block = mass x acceleration
Step 3: Determine the weight of each block:
The weight (mg) of an object is equal to its mass multiplied by the acceleration due to gravity (9.8 m/s^2)
Weight of 90.0 kg block = 90.0 kg x 9.8 m/s^2 = 882.0 N
Weight of 40.0 kg block = 40.0 kg x 9.8 m/s^2 = 392.0 N
Step 4: Write the equations of motion for each block using the weight values:
For the 90.0 kg block:
Tension in the rope - 882.0 N = 90.0 kg x acceleration (Equation 1)
For the 40.0 kg block:
Tension in the rope - 392.0 N = 40.0 kg x acceleration (Equation 2)
Step 5: Solve the system of equations simultaneously to find the values of acceleration and tension:
To eliminate the tension variable, we can subtract Equation 2 from Equation 1:
(Tension in the rope - 882.0 N) - (Tension in the rope - 392.0 N) = (90.0 kg x acceleration) - (40.0 kg x acceleration)
Simplifying the equation:
490.0 N = 50.0 kg x acceleration
Now we can solve for the acceleration:
acceleration = 490.0 N / 50.0 kg = 9.8 m/s^2
The acceleration of the larger block is 9.8 m/s^2.
Step 6: Calculate the tension in the rope:
Substitute the value of the acceleration into one of the equations:
Tension in the rope - 882.0 N = 90.0 kg x 9.8 m/s^2
Simplifying the equation:
Tension in the rope = 882.0 N + 882.0 N = 1764.0 N
The tension in the rope is 1764.0 N.