Note: The left-hand mass is equal to the sum of the other two masses.
A massless pulley is attached to the ceiling,in a uniform gravitational field, and rotates with no friction about its pivot. Another massless pulley is attached to the end of the
cord on the right and rotates with no friction about its pivot.
The acceleration of gravity is 9.8 m/s2 .
---pulley 1
---|------|
---|------|
---|----pulley 2
---|-----|------|
---|-----|------|
-46 kg--36 kg--10 kg
What is the magnitude of the acceleration of the 36 kg mass?
Answer in units of m/s2
To find the magnitude of the acceleration of the 36 kg mass, we can use Newton's second law of motion.
First, let's assign directions to the motion. Let's say that the mass with 10 kg is moving up, and the mass with 46 kg is moving down. Therefore, the 36 kg mass will also move down.
Now, we can write down the equations of motion for each mass:
For the 10 kg mass:
T - m1g = m1a (equation 1)
For the 36 kg mass:
T - m2g = m2a (equation 2)
For the 46 kg mass:
m3g - T = m3a (equation 3)
In these equations, T represents the tension in the cord, m1, m2, and m3 represent the masses, g represents the acceleration due to gravity, and a represents the acceleration.
We can see from the diagram that the tension in the cord is the same for all masses. Therefore, we can rewrite equation 1 and equation 3 as:
T = m1g + m1a (equation 1')
T = m3g - m3a (equation 3')
Now, we can solve this system of equations to find the acceleration of the 36 kg mass.
From equation 2, we have:
T - m2g = m2a
Substituting the value of T from equation 1' and equation 3', we get:
m1g + m1a - m2g = m2a
m1g = (m2 + m1 - m2)a
a = m1g / (m1 + m2)
Substituting the given values:
a = (36 kg)(9.8 m/s^2) / (36 kg + 10 kg)
= 352.8 kg·m/s^2 / 46 kg
≈ 7.67 m/s^2
Therefore, the magnitude of the acceleration of the 36 kg mass is approximately 7.67 m/s^2.