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Freebody Diagrams

Ball Freebody

Use of a simple accelerometer

Our second freebody shows the ball has two forces acting on it Tension force, T and Gravitational force, Gb.

Our idealized system

By: Noah Whitney

Newton's 2nd Law illustrates the relationship between net force and acceleration:

F=ma (net force = mass times acceleration)

We can draw two freebody diagrams: one of our system ot find the net force accelerating it and a second one of our ball to examine the forces acting on it and calculate it's acceleration.

Our system freebody finds only three external forces acting on the system: the gravitational force on m1, the gravitational force on m2 and the normal force on m2. Since it is on a horizontal surface the normal force and gravitational force acting on m2 are equal and opposite. Therefore the net force on the system is equal to the gravitational force exerted on m1. Using Newtons laws of motion we find that G1=m1×g and the acceleration, a=(m1×g)/(m1+m2).

To simplify our problem we will make two assumptions about our system:

  • We assume all friction acting on the system is negligible
  • We assume the strings holding our sysytem together and the ball pictured have negligible mass

Formulas and laws

The following formulas and laws are critical to understanding our accelerometer:

  • Newton's 1st law: An object in motion will remain in motion and an object at rest will remain at rest unless acted upon by an unbalanced force

  • Newton's second law: F=ma

  • Right triangle definition of tan: tan = opp/adj

Vector Decomposition

Conclusion

We can decompose vector t into it's x and y components (tx and ty). We can also deduce that since the ball is not moving vertically the sum of forces ty and g acting on it vertically is equal to zero (Newton's 1st law). Since ty and g are working in opposite directions, ty=g.

Finally, we know tx=a since the system is only accelerating in the x direction. Using some basic trigonometry we can write a formula relating the two to ⁡angle θ, g×tan⁡θ=a.

A Single System

With the formula g×tan⁡θ=a, we can calculate the acceleration based solely on angle theta (g is a constant on earth's surface). If we input it into the other formula we found (a=(m1×g)/(m1+m2)) it can also tell us the ratio between masses m1 and m2. When m1 is far greater than m2 the acceleration will approach g and ⁡θ will approach 45 degrees which is tan(1). When m1 is much less than m2 the acceleration will approach 0 and tan⁡θ will approach 0 degrees which is tan(0).

Problem statement: What can angle ⁡θ tell us about the accleration of the system and masses within it?

Our system is comprised of two masses, the platform (m2) and a second mass (m1), as well as a ball hanging like a pendulum. These can be treated as a single system with a singular acceleration and velocity since they are connected inelastically. If they were accelerating at different rates they would seperate, but we can observe that this is not the case.

References

I would consider introductory trigonometry and Newton's laws of motion to be considered universal knowledge thus making referencing unnecessary but just in case...

Newton's laws and vector decomposition: Chapters 4 and 5 of Sears and Zemansky’s University Physics, 13th edition, Young and Freedman, Pearson education

Right triangle definition of tan:

Lumen learning, https://courses.lumenlearning.com/boundless-algebra/chapter/trigonometry-and-right-triangles/

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