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Calculate the Acceleration of Two Blocks Connected by A String | Half Atwood Machine
INTEGRAL PHYSICS
Overview
This video explains how to calculate the acceleration and tension in a simplified "half Atwood machine" system. It involves a block on a horizontal surface connected by a string over a pulley to a hanging mass. The explanation focuses on drawing free-body diagrams for each mass, applying Newton's second law, and solving the resulting system of equations. The video emphasizes setting up a consistent direction of motion for both blocks and derives general formulas for acceleration and tension in terms of the masses and gravitational acceleration, assuming no friction or pulley mass.
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Chapters
- The problem involves two masses connected by a string over a pulley, a simplified version of the Atwood machine.
- The goal is to find the acceleration of both masses and the tension in the string.
- This specific scenario assumes no friction on the horizontal surface and no friction or mass in the pulley.
Understanding this basic setup is crucial for grasping Newton's laws and forms the foundation for more complex mechanics problems.
- For the block on the table (m1), forces are gravity (down), the normal force (up), and tension (right).
- Gravity on m1 (m1g) is balanced by the normal force because there's no vertical acceleration.
- For the hanging block (m2), forces are gravity (down) and tension (up).
- Crucially, the tension in the string is NOT equal to the weight of the hanging block (m2g) because the system is accelerating.
Accurately identifying and representing all forces acting on each object is the essential first step in applying Newton's laws.
Drawing arrows to represent gravity, normal force, and tension acting on each of the two blocks.
- Since the blocks are connected, they have the same magnitude of acceleration but in opposite directions (right for m1, down for m2).
- To apply Newton's second law consistently, define a positive direction of motion.
- The chosen positive direction is to the right for the block on the table and downward for the hanging block.
Establishing a consistent coordinate system is vital for correctly setting up equations when dealing with vector quantities like force and acceleration in a coupled system.
Designating 'right' as positive for the block on the table and 'down' as positive for the hanging block, ensuring both accelerations have the same sign in the equations.
- Apply Newton's second law (ΣF = ma) to each block separately.
- For the block on the table (m1): Tension (T) = m1 * a (since net force is only tension in the defined positive direction).
- For the hanging block (m2): m2g - Tension (T) = m2 * a (since gravity is in the positive direction and tension is in the negative direction).
- Solve the system of two equations with two unknowns (a and T) using substitution.
This is where the physical principles are translated into mathematical relationships, allowing for the quantitative determination of the system's motion.
Substituting the expression for T from the first equation (T = m1a) into the second equation (m2g - T = m2a) to solve for 'a'.
- Solving the system yields an equation for acceleration: a = (m2 * g) / (m1 + m2).
- Substituting this acceleration back into the tension equation yields: T = (m1 * m2 * g) / (m1 + m2).
- These formulas allow calculation of acceleration and tension given the masses and gravitational acceleration.
These derived formulas provide a general solution that can be applied to any specific set of masses in this idealized scenario, demonstrating the power of symbolic physics.
Plugging in numerical values for m1, m2, and g into the derived formula for 'a' to find the system's acceleration.
Key takeaways
- Free-body diagrams are essential for visualizing and analyzing forces on individual objects.
- In a connected system, masses accelerate together, requiring a consistent definition of positive direction for all parts.
- Tension in a string connecting accelerating masses is not necessarily equal to the weight of any hanging portion.
- Newton's second law (ΣF = ma) must be applied to each mass individually.
- Solving a system of equations derived from Newton's second law allows for the determination of unknown variables like acceleration and tension.
- The derived formulas for acceleration and tension are valid only under the stated assumptions (no friction, massless pulley).
Key terms
Half Atwood MachineNewton's Second LawFree-Body DiagramTensionAccelerationNormal ForceGravitySystem of Equations
Test your understanding
- What are the forces acting on the block resting on the table, and how do they relate if the block is not accelerating vertically?
- Why is it important to define a consistent direction of motion when analyzing the acceleration of connected masses?
- How does the tension in the string of a half Atwood machine differ from the weight of the hanging mass when the system is accelerating?
- What is the general formula for the acceleration of the two blocks in this system, and what assumptions does it rely on?
- How can you use the derived formula for acceleration to find the tension in the connecting string?