What is the formula for tension at various angles in a constrained mass system?

In summary, the conversation discusses a problem involving a mass constrained to move along a linear track with a magnet attached to it. A person holds another magnet near the first magnet, causing the mass to accelerate. The angle between the mass and the horizontal axis is denoted as theta, and the person wants to know the tension force acting on the mass at different angles. The solution involves using trigonometry and decomposing vectors into components.
  • #1
e2m2a
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[No template as this thread was moved to Homework from the general physics forum]

I have been struggling with this problem for years and never have found an answer to it anywhere on the web or in textbooks. And I can’t derive the formula for it.

Suppose we have a mass designated as ##m_2##. This mass is constrained to move with one degree of freedom along a linear track. We assume no friction between ##m_2## and the track. The track is rigidly attached to the earth. We define the motion of ##m_2## to be along the y-axis.

There is a near-massless aluminum rod attached at one end to ##m_2##. At the other end of the rod is attached a magnet with mass ##m_1##. A person holds another magnet near ##m_1##, such that there is a magnetic force experienced by ##m_1## in the radially outward direction. We designate this force on ##m_1## as ##f_{mag}##. Because of this ##f_{mag}##, ##m_2## accelerates in the positive y-direction along the linear track.

The person hovers over the Earth in some kind of spacecraft or whatever, so there is no contact forces between the Earth and the bottom of the shoes of the person. And with this spacecraft , the person is able to keep the distance between the two magnets constant and keeps up with the accelerating ##m_2##, such that ##f_{mag}## remains constant.

We define the angle ##\theta## as the angle between the rod and the horizontal x-axis.

Here is my conundrum. I want to know the tension force, designated ##f_{tension}## acting on ##m_2## for all angles of the rod.

I know for the special case where ##\theta## is equal to 90 degrees, it would simply be:

$$f_{tension} = \frac {m_2} {m_1 + m_2} f_{mag}$$

And for ##\theta## equal to 0 degrees, it would be:

$$f_{tension} = \frac {m_{earth}} {m_1 + m_{earth}} f_{mag}$$

Where, ##m_{earth}## is the combined mass of the Earth and ##m_2##.

But I cannot figure out the formula for ##f_{tension}## for any angle ##\theta## between 0 and 90 degrees. Can someone help me out with this?
 
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  • #2
I can't help you with the problem but I strongly suggest that for clarity you draw a diagram of exactly what you mean.
 
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  • #3
Here is an attached drawing.
 

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  • #4
Two concepts that you'll need here:
1) Trigonometry. Are you familiar with the ##\sin## and ##cos## functions?
2) Decomposing vectors into components.
 
  • #5
Nugatory said:
Two concepts that you'll need here:
1) Trigonometry. Are you familiar with the ##\sin## and ##cos## functions?
2) Decomposing vectors into components.

I am familiar with both. However, I can't conceptualize and put it into mathematics how the tension force changes as a function of the angle. For example, let's say ##\theta## is 45 degrees. It seems to me that the tension force would by a "hybrid" force, a function of ##m_1## interacting with ##m_2## and ##m_{earth}## along the x-axis, and ##m_1## simultaneously interacting with just ##m_2## along the y-axis. Because of this, I can't decide how to set up the trig equations.
 
  • #6
Have you drawn free body diagrams for m1 and m2?
 

Related to What is the formula for tension at various angles in a constrained mass system?

What is tension at various angles?

Tension at various angles refers to the force applied to an object at different angles. This force is typically measured in Newtons (N) and can vary depending on the angle of application.

How is tension at various angles calculated?

The formula for calculating tension at various angles is T = F * cosθ, where T is the tension, F is the force applied, and θ is the angle of application. This formula is derived from the laws of trigonometry and can be used to determine the tension at any given angle.

Why is tension at various angles important?

Tension at various angles is important because it affects the stability and strength of structures and objects. Understanding the tension at different angles is crucial in engineering and construction to ensure the safety and durability of buildings and other structures.

What factors can affect tension at various angles?

The tension at various angles can be affected by factors such as the magnitude of the applied force, the angle of application, and the properties of the object or structure. Other environmental factors, such as wind or temperature, can also impact tension at various angles.

How can tension at various angles be controlled?

Tension at various angles can be controlled by adjusting the magnitude and direction of the applied force, as well as by using different materials or structural designs. Proper maintenance and regular inspections can also help to prevent excessive tension and ensure the stability of structures.

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