What if the pulley has mass?

If the pulley has mass or a moment of inertia, the tensions on either side of the pulley are no longer equal. You must write a torque equation for the pulley ($\Sigma au = I \alpha$) and relate angular acceleration to linear acceleration ($\alpha = a/R$).

What if there is friction on a horizontal surface?

In a modified Atwood machine where one mass rests on a rough horizontal surface, include the kinetic friction force ($f_k = \mu_k N$) in the horizontal $\Sigma F = ma$ equation. It opposes the direction of motion.

How do I know which way the system will accelerate?

Guess a direction. If your final calculated acceleration is negative, the system accelerates in the opposite direction of your initial guess. The magnitude remains correct as long as friction is not involved (static friction direction depends on the actual tendency of motion).

How to solve a pulley problem with two masses

To solve a two-mass pulley problem, isolate each mass and apply Newton's second law simultaneously. The masses are linked by a shared string, meaning they share the same magnitude of acceleration and the same string tension. This method applies to ideal systems where two objects are connected by an inextensible string passing over a massless, frictionless pulley, such as an Atwood machine or a modified Atwood machine.

The setup

Isolate the system into two distinct objects: mass 1 ( $m_1$ ) and mass 2 ( $m_2$ ). Assume the string is inextensible and the pulley is massless and frictionless. Let the tension in the string be $T$ and the magnitude of acceleration be $a$ . Define a unified coordinate system that follows the motion of the string. For example, if $m_2$ accelerates downward, define downward as positive for $m_2$ and upward as positive for $m_1$ .

The steps

Draw a free-body diagram (FBD) for each mass. Include gravity ( $mg$ ), normal forces, friction, and the tension force ( $T$ ). 2. Choose a sign convention that bends around the pulley. The direction of expected acceleration should be positive for both masses. 3. Write Newton's second law ( $\Sigma F = ma$ ) for each mass using the chosen sign convention. 4. Add the two resulting equations. The tension $T$ will cancel out, allowing you to solve for the acceleration $a$ . 5. Substitute $a$ back into either original equation to solve for $T$ .

Checking the result

Test the limit cases of your algebraic answer. If $m_1 = m_2$ in a standard vertical Atwood machine, the acceleration $a$ must evaluate to zero. If $m_1$ is zero, the acceleration of $m_2$ must evaluate to $g$ (free fall). Verify that the tension $T$ is less than the weight of the heavier mass but greater than the weight of the lighter mass.

Common errors

A frequent error is assigning opposite signs to the acceleration of the two masses relative to the string's motion. If you define 'up' as positive for both masses in an Atwood machine, you must use $a$ for one mass and $-a$ for the other. Another common mistake is assuming the tension $T$ is equal to $mg$ for either mass. Tension only equals weight if the system is in equilibrium ( $a=0$ ).

Worked example

Two masses, $m_1 = 3.0$ kg and $m_2 = 5.0$ kg, hang from the ends of an inextensible string over a massless, frictionless pulley (a standard Atwood machine). Calculate the acceleration of the system and the tension in the string. Use $g = 9.8$ m/s $^2$ .

Let the direction of motion for the heavier mass ( $m_2$ ) be positive. Downward is positive for $m_2$ , and upward is positive for $m_1$ . For $m_1$ : $\Sigma F_1 = T - m_1 g = m_1 a$ . For $m_2$ : $\Sigma F_2 = m_2 g - T = m_2 a$ . Add the two equations: $(T - m_1 g) + (m_2 g - T) = m_1 a + m_2 a$ . This simplifies to: $m_2 g - m_1 g = (m_1 + m_2) a$ . Solve for $a$ : $a = g (m_2 - m_1) / (m_1 + m_2)$ . Substitute the values: $a = 9.8 (5.0 - 3.0) / (3.0 + 5.0) = 9.8 (2.0) / 8.0 = 2.45$ m/s $^2$ . To find $T$ , substitute $a$ into the equation for $m_1$ : $T - 3.0(9.8) = 3.0(2.45)$ . $T - 29.4 = 7.35$ . $T = 36.75$ N.

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References: University Physics with Modern Physics (Young & Freedman) - Chapter 5 · OpenStax University Physics Volume 1 - Chapter 6 · Fundamentals of Physics (Halliday, Resnick, Walker) - Chapter 5