In a device known as an Atwood machine, two masses m1=10kg and m2=20kg are connected by rope over a frictionless pulley. What is the acceleration of each mass if the rope is massless?

To find the acceleration of each mass in an Atwood machine, we can use Newton's second law of motion. The net force acting on each mass is equal to the product of its mass and its acceleration.

Let's assume m1 is the smaller mass and m2 is the larger mass. In this system, the smaller mass will experience an upward force and the larger mass will experience a downward force due to gravity. The difference in these forces will cause the system to accelerate.

To start solving the problem, we need to calculate the net force on each mass:

For m1:
The downward force due to gravity is given by F = m1 * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

For m2:
The upward force due to gravity is given by F = m2 * g.

Since the rope is assumed to be massless and there is no friction, the tensions in the rope on either side of the pulley are equal.

Now, let's calculate the net force acting on each mass:

For m1:
Net force on m1 = Tension - m1 * g

For m2:
Net force on m2 = m2 * g - Tension

Since the accelerations of both masses are equal and opposite in direction, the magnitudes of the net force on each mass must be equal. Therefore:

Tension - m1 * g = m2 * g - Tension

Now, let's rearrange the equation:

2 * Tension = m2 * g + m1 * g

Tension = (m2 * g + m1 * g) / 2

Let's substitute the given values:

m1 = 10 kg
m2 = 20 kg
g = 9.8 m/s^2

Tension = (20 kg * 9.8 m/s^2 + 10 kg * 9.8 m/s^2) / 2

Tension = (196 kg m/s^2 + 98 kg m/s^2) / 2

Tension = 294 kg m/s^2 / 2

Tension = 147 kg m/s^2

Now, we have the value of tension in the rope. We can use this to calculate the acceleration of each mass.

For m1:
Net force on m1 = Tension - m1 * g = 147 kg m/s^2 - 10 kg * 9.8 m/s^2 = 147 kg m/s^2 - 98 kg m/s^2 = 49 kg m/s^2

For m2:
Net force on m2 = m2 * g - Tension = 20 kg * 9.8 m/s^2 - 147 kg m/s^2 = 196 kg m/s^2 - 147 kg m/s^2 = 49 kg m/s^2

Finally, the acceleration of each mass is given by:

Acceleration = Net force / mass

For m1:
Acceleration = 49 kg m/s^2 / 10 kg = 4.9 m/s^2

For m2:
Acceleration = 49 kg m/s^2 / 20 kg = 2.45 m/s^2

Therefore, the acceleration of m1 is 4.9 m/s^2, and the acceleration of m2 is 2.45 m/s^2.