An Atwood's machine consists of two crates of different masses. If the pulley is frictionless and
massless along with a string of negligible mass, find the acceleration of the crates and the
tension in the string
assume the weight of the hanging down mass to be M2, with the other mass is horizonal movement, and mass M1
net force=m2*g=totalmass*a
m2*g=(M1+M3)a
now tension between them. the tension is pulling mass1
tension=m1*a
or, tension is holding back M2 and
m2g-tension=M2a
either equation willwork, and you should be able to confirm these equations with individual free body diagrams, and UNDERSTAND those (Hint: you will likely see it on a test)
To find the acceleration of the crates and the tension in the string in an Atwood's machine setup, you need to apply Newton's second law of motion. Here's how you can do it step by step:
1. Identify the masses of the two crates: Let's call the masses of the crates m1 and m2, where m1 is the larger mass and m2 is the smaller mass.
2. Set up your coordinate system: Choose a convenient direction as positive and assign the direction of motion for one of the crates as positive. This convention will help you correctly assign signs to the accelerations.
3. Apply Newton's second law to each crate: The net force acting on each crate is equal to its mass multiplied by its acceleration.
For crate 1 (m1):
- The force pulling downward is m1g (weight of crate 1).
- The force pulling upward is T (tension in the string).
- The net force is equal to m1 multiplied by the acceleration.
So, m1g - T = m1a
For crate 2 (m2):
- The force pulling downward is m2g (weight of crate 2).
- The force pulling upward is T (tension in the string).
- The net force is equal to m2 multiplied by the acceleration.
So, T - m2g = m2a
4. Solve the simultaneous equations: Combine the two equations to eliminate T and solve for the acceleration, a.
(m1g - T) / m1 = (T - m2g) / m2
Simplify the equation to: m1g - T = (m1 + m2) * a
Now, solve for a: a = (m1g - T) / (m1 + m2)
5. Substitute the value of acceleration: Use the value of acceleration you obtained in step 4 to calculate the tension in the string.
T = m2g + m2 * a
6. Calculate the acceleration and tension: Substitute the given values of m1, m2, and g into the equations to find the acceleration and tension.
Once you have solved these equations, you will obtain the values of the acceleration (a) of the crates and the tension (T) in the string for the given masses of the crates.