Solve Vibration Problem: Find m/m1 Ratio for Stationary Point A

[gholamghar]

Homework Statement

in this vibration system find ratio of (m/m1) so that (point A) would not move and will stay stationary.
point A distance from left end of spring is 0.25 of total length of spring.
all surfaces are frictionless.

the problem is in indie numbers:

The Attempt at a Solution

here is the answer but i can not find out how the problem is solved if any
can give me an explanation about the answer i really would appreciate it.

 

[KataKoniK]

The ratio of (m/m1) can be found using the following steps:

1. Draw a free body diagram of the system, with point A being at a distance of 0.25L from the left end of the spring.
2. Apply Newton's Second Law (F=ma) to the system, considering the forces acting on both masses m and m1.
3. Since the surfaces are frictionless, the only forces acting on the masses are the spring forces and the weight of the masses.
4. Set up equations for the forces acting on each mass, using the distance of point A and the total length of the spring.
5. Equate the spring forces and the weight forces for each mass, and solve for the ratio (m/m1).
6. Simplify the equation and solve for the ratio (m/m1).

The final answer will depend on the specific values of the masses, the spring constant, and the length of the spring. However, the general approach is to analyze the forces acting on the system and set up equations to solve for the unknown ratio (m/m1).

 

[Mene]

I would approach this vibration problem by first understanding the basic principles of vibration and equilibrium. Vibration is a result of a dynamic equilibrium between the forces acting on a system. In this case, the forces acting on the system are the weight of the mass m1 and the spring force.

To solve this problem, we need to find the ratio of m/m1 so that point A remains stationary. This means that the forces acting on point A must be balanced, resulting in a net force of zero. In other words, the weight of m1 must be equal to the spring force acting on point A.

To find the spring force, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement. In this case, the displacement of the spring is 0.25 of the total length, so the spring force would be 0.25 times the spring constant.

Next, we can set up an equation to represent the forces acting on point A:

m1g = (0.25k)(0.25L)

Where m1 is the mass of the object, g is the acceleration due to gravity, k is the spring constant, and L is the total length of the spring.

Solving for m/m1, we get:

m/m1 = (0.25k)(0.25L)/m1g

By rearranging the equation, we can see that the ratio m/m1 is dependent on the spring constant, the total length of the spring, and the acceleration due to gravity. This means that the ratio can be adjusted by changing any of these variables.

In conclusion, to solve this vibration problem and find the ratio of m/m1 for point A to remain stationary, we need to understand the principles of vibration and equilibrium, use Hooke's Law to calculate the spring force, and set up an equation to represent the forces acting on point A. By manipulating the equation, we can determine the ratio of m/m1.