To solve the problem of two masses connected by a spring on a smooth horizontal surface, we need to analyze the forces and motions involved when the spring is initially stretched and then compressed. Let's break it down step by step.
Understanding the System
We have two masses, m1 and m2, connected by a spring with a spring constant k. Initially, the spring is stretched by a distance d. When the system is released from rest, the spring will exert forces on both masses as it tries to return to its equilibrium position.
Initial Conditions
When the spring is stretched by distance d, it stores potential energy given by the formula:
- Potential Energy (PE) = (1/2) * k * d²
This potential energy will be converted into kinetic energy as the spring returns to its natural length.
Motion of the Masses
As the spring returns to its equilibrium position, both masses will start moving towards each other. Since the surface is smooth, we can ignore friction, and the only forces acting on the masses are due to the spring.
Equations of Motion
When the spring is compressed by a distance d, it will exert a force on both masses. The force exerted by the spring when compressed is given by Hooke's Law:
Here, x is the distance the spring is compressed. When the spring is compressed by distance d, the force acting on each mass will be equal and opposite, causing them to accelerate towards each other.
Finding the Distance Moved by Each Mass
Let’s denote the distance moved by mass m1 as x1 and the distance moved by mass m2 as x2. Since the spring is symmetric and both masses are connected, we can assume that they will move towards each other equally when the spring is compressed. Therefore, we can say:
When the spring is compressed by distance d, the total distance moved by both masses will be:
- Total Distance = x1 + x2 = x + x = 2x
Conservation of Energy
Using the conservation of energy principle, the initial potential energy stored in the spring when stretched will equal the kinetic energy when the spring is compressed:
As the spring compresses, the potential energy is converted back into kinetic energy. The total energy in the system remains constant, allowing us to set up the equation:
- (1/2) * k * d² = (1/2) * m1 * v1² + (1/2) * m2 * v2²
Where v1 and v2 are the velocities of m1 and m2, respectively. Since they start from rest and move towards each other, we can relate their velocities to the distances they move.
Final Calculation
When the spring is compressed by distance d, the distance moved by each mass can be derived from the energy equations and the relationship between their velocities. Assuming equal masses and equal distances moved, we can conclude:
- Distance moved by each mass = d/2
Thus, when the spring is compressed by distance d, each mass moves a distance of d/2 towards each other. Therefore, the total distance moved by the two masses when the spring is compressed by distance d is:
- Total Distance = d/2 + d/2 = d
In summary, when the spring is compressed by a distance d, the two masses m1 and m2 will each move a distance of d/2 towards each other, resulting in a total movement of d. This analysis illustrates the interplay of forces, energy conservation, and the dynamics of connected systems.
