6.4:
Second Law: Motion under Same Acceleration
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.
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 moves at the same acceleration as that of the tractor.
Consider Block 1 of a certain mass on a frictionless, horizontal surface being pulled by a light string that passes over a frictionless and massless pulley. The other end of the string is connected to Block 2 of a different mass over the edge of the surface. Here, Newton's second law is applied to find the acceleration and tension in the string. To do this, a free-body diagram for each mass is drawn separately, and the forces acting on them are identified.
The forces on the block lying on the horizontal surface are gravitational force, the contact force of the surface, and the tension in the string. The hanging block is subjected to gravitational force and the string tension. Note that the tension is the same throughout the string. All forces are either horizontal or vertical, so horizontal/vertical coordinate systems for both objects are used to write component equations for each block. Block 1 has balanced vertical forces, so only the equation relating to the x-components is considered. There are no horizontal forces on Block 2, so only the y-component equation is considered. When Block 1 moves to the right, Block 2 travels an equal distance downward; therefore, they have the same acceleration.
If both these blocks are hung vertically using a massless string over a frictionless pulley, then the acceleration remains the same as one moves upward and the other downward.
This text is adapted from Openstax, University Physics Volume 1, Section 6.1: Solving Problems with Newton's Laws.