Final answer:
To calculate the tension in the string of an Atwood machine, one must use Newton's second law, considering the forces and acceleration of the masses involved. An equation involving the masses of the blocks and gravity can be derived to find the acceleration, which is then utilized to find the tension.
Step-by-step explanation:
To find the tension in the string of an Atwood machine, we must apply Newton's second law to the system. This law states that the sum of the forces on an object is equal to the product of the mass of the object and its acceleration (F = ma). In an Atwood machine, two masses are connected by a string over a pulley, and we assume the masses of the string and the pulley are negligible. To find an equation for the acceleration of the blocks, we take the difference in weight (mg) of the two blocks (where m is the mass and g is the acceleration due to gravity) and divide it by the total mass of the two blocks. This gives us:
a = (m2 - m1)g / (m1 + m2)
To find the tension in the string, we analyze the forces on one of the masses (since the tension throughout the string is the same if the pulley is massless and frictionless). For mass m1, the net force is the tension upwards minus the weight (m1g) downwards:
T - m1g = m1a
Substitute the acceleration (a) we previously found into this equation:
T = m1(m2 - m1)g / (m1 + m2) + m1g
For example, if block 1 has mass 2.00 kg and block 2 has mass 4.00 kg:
a = (4.00 kg - 2.00 kg) * 9.8 m/s2 / (2.00 kg + 4.00 kg)
a = 3.27 m/s2
Then, we can calculate the tension:
T = 2.00 kg * ((4.00 kg - 2.00 kg) * 9.8 m/s2 / (2.00 kg + 4.00 kg)) + 2.00 kg * 9.8 m/s2
T = 29.4 N