Three blocks of masses m1 = 4 kg, m2 = 2 kg, m3 = 4 kg are connected with ideal strings passing over a smooth massless pulley. What will be the acceleration of the blocks?

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Understand the Problem

The question is asking to find the acceleration of three connected blocks of different masses that are influenced by the force of gravity and tension in strings, indicating a problem in classical mechanics involving Newton's laws.

Answer

The acceleration of the blocks is $3.33 \, \text{m/s}^2$.
Answer for screen readers

The acceleration of the blocks is approximately $3.33, \text{m/s}^2$.

Steps to Solve

  1. Identify the Forces Acting on Each Block

    The system has three blocks connected by ideal strings passing over a pulley.

    • For $m_1 = 4, \text{kg}$ (on the left): The force due to gravity is $F_g = m_1 g = 4 \times 10 = 40, \text{N}$, acting downwards.
    • For $m_2 = 2, \text{kg}$: The force due to gravity is $F_g = m_2 g = 2 \times 10 = 20, \text{N}$, acting downwards.
    • For $m_3 = 4, \text{kg}$ (on the right): The force due to gravity is the same as $m_1$, $F_g = 40, \text{N}$, acting downwards.
  2. Apply Newton's Second Law

    Using Newton's second law, we write the equations for the blocks based on the forces acting on them.

    • For $m_1$:
      $$ T - m_1 g = -m_1 a $$ (where $T$ is the tension and $a$ is the acceleration).

    • For $m_2$:
      $$ T - m_2 g = -m_2 a $$

    • For $m_3$:
      $$ T - m_3 g = -m_3 a $$

    This gives us three equations.

  3. Solve for Tension and Acceleration

    Combine the equations to eliminate $T$ and solve for the acceleration $a$ of the system.

    • From the second block's equation: $$ T = m_2 g + m_2 a $$ Substitute into the first block's equation:

    $$ m_2 g + m_2 a - m_1 g = -m_1 a $$

    Rearranging yields: $$ m_2 g - m_1 g = (m_1 + m_2) a $$

    Substitute the masses and solve for $a$.

  4. Calculate the Result

    Substitute the values into the equation: $$ (20 - 40) = (4 + 2) a $$ Simplifying gives: $$ -20 = 6a $$ Thus: $$ a = -\frac{20}{6} \approx -3.33 , \text{m/s}^2. $$

  5. Check for Direction and Units

    Since acceleration cannot be negative in this context, we conclude the acceleration's magnitude is: $$ a = 3.33 , \text{m/s}^2. $$

The acceleration of the blocks is approximately $3.33, \text{m/s}^2$.

More Information

This problem illustrates the principles of Newton's laws of motion and the effects of connected systems. The calculated acceleration indicates how quickly the masses accelerate due to the forces acting on them.

Tips

  • Misidentifying the direction of tension and gravity. Remember, tension always acts opposite to the direction of the mass's weight.
  • Failing to correctly account for the net forces acting in the same direction.
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