6.4
Second Law: Motion under Same Acceleration
Newton's second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart…
Bodies moving with the same acceleration obey Newton's second law. Consider a car towing another identical car with a massless inextensible rope. The rope applies equal and opposite force on both cars. Then, why should the car accelerate?
To answer this, a free-body diagram of each is drawn. Since the vertical forces are balanced, the net force acting on each is obtained using the horizontal force components.
Now, using the fact that both the cars have the same mass and acceleration, the equations are equated to get the force causing the cars to accelerate.
Consider another example of a massless, inextensible string hanging two unequal masses over a frictionless pulley. When the system is released, the heavier mass lifts the lighter one, and they both move with the same acceleration.
Using free-body diagrams and Newton's second law, an equation for the net force on each mass can be derived. On subtracting these equations and rearranging them, the value of the acceleration of the masses is obtained.
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Q1: Why do two cars connected by a rope accelerate together?
When a car tows another identical car with a massless inextensible rope, both experience the same acceleration because the rope applies equal and opposite forces on each. Using free-body diagrams, the net horizontal force on each car equals its mass times acceleration. Since both cars have identical mass and acceleration, they move together as a system.
Q2: How do you find acceleration when unequal masses hang over a pulley?
Draw free-body diagrams for each mass and apply Newton's second law separately. The heavier mass pulls the lighter one upward while descending. Both masses move with the same acceleration despite their different weights. Subtract the force equations and rearrange to solve for the acceleration value.
Q3: What role does tension play in a string connecting two blocks?
Tension is the force transmitted through a massless string connecting two blocks. The tension remains constant throughout the string and acts horizontally on a block on a surface and vertically on a hanging block. Tension pulls each block in opposite directions, enabling them to move with the same acceleration.
Q4: Why do blocks on a horizontal surface and hanging blocks have equal acceleration?
When one block moves horizontally on a frictionless surface and another hangs vertically, connected by a string over a pulley, they travel equal distances in equal times. This constraint means both blocks must have the same acceleration magnitude. Newton's second law applied to each block confirms this kinematic relationship.
Q5: How do free-body diagrams help solve problems with same acceleration?
Free-body diagrams isolate each object and show all forces acting on it. For objects with the same acceleration, you write component equations for horizontal and vertical forces separately. Balanced forces in one direction simplify the analysis, allowing you to focus on net forces causing acceleration.
Q6: What happens to tension when two blocks of different masses hang vertically?
When unequal masses hang vertically over a frictionless pulley, the tension in the string is less than the weight of the heavier mass but greater than the weight of the lighter mass. The tension accelerates the lighter mass upward and reduces the net downward force on the heavier mass, allowing both to accelerate together.
Q7: Why are vertical and horizontal forces analyzed separately in these problems?
Forces perpendicular to each other do not affect one another. On a horizontal surface, vertical forces balance, so only horizontal components determine acceleration. For hanging blocks, only vertical forces matter. Separating components simplifies calculations and reveals which forces actually cause the observed acceleration.