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Statics & Dynamics: Tips & Tricks

Statics & Dynamics: Tips & Tricks

Problem Solving Guide

Algorithm 

  1. Read and re-read the whole problem carefully.
  2. Visualize the scenario. Mentally try to understand what the rigid body is doing and what forces are acting on it along with the location of the forces on the body.
  3. Draw a physical representation of the scenario, this may include a FBD and/or an e-FBD.
    1. Draw all forces acting on the object at the known locations.  (Helpful hint:  Draw a FBD first.  All the forces on the FBD will show up on the e-FBD, no more, no less.)
    2. Define a coordinate system on your e-FBD.  Conventional coordinate system is ccw(+) and cw(-).  
    3. Label a location that represents the axis that you would sum torques around.  The location of this axis can be placed anywhere, but defining it to be at the location of unknown forces is a good way to eliminate the torque due to the unknown force.
    4. Specify the moment arms for all forces acting on the e-FBD.
  4. Identify all the known and unknown terms.
  5. Use Newton's rotational version of the second law and sum the torques around the axis you chose.
    1. Identify if the object is in rotational equilibrium or not to determine the value of the angular acceleration.
    2. If the angular acceleration is not zero, determine the moment of inertia of the object around the axis you chose.
  6. Determine the number of unknowns from your rotational version of Newton's second law equation.  If more than 1, perhaps choose a different axis to sum the torques around, or draw a regular FBD to build more equations.
  7. Carry out the algebraic process of solving the equation(s).
  8. Evaluate your answer, make sure the units are correct and the results are within reason.

Misconceptions & Mistakes

  • The axis you choose to sum the torques around does not have to be the actual pivot point that the object would rotate around.
  • When calculating torque, do not forget to add a positive or negative based off of the direction that the force wants to rotate the rigid body about the axis you chose.
  • An object does not have a single moment of inertia.  The moment of inertia depends on the axis that the object would rotate around.  Thus objects can have many moments of inertia for different axes.
  • Torque has the same dimensions of energy, but torque is not a form of energy.  Torque is a vector, and energy is a scalar

Pro Tips

  • For any force that is not parallel or perpendicular to the moment arm, take the time to draw a vector operation to determine the correct angle between the force and moment arm.
  • If you are having troubles identifying forces for an e-FBD, draw a regular FBD first.  The same forces that show up on a regular FBD for an object show up on an e-FBD for the same object. In addition, often-times using both a translation analysis and a rotational analysis together will help expedite the process of solving the problem.
  • Identify all the known and unknown forces on your e-FBD before you define an axis to sum the torques around.  Once you know which forces are unknowns, try defining your axis through the location of the unknown forces to eliminate the unknown force in the final equation derived from summing the torques.
  • If there is a torque which an unknown direction, just guess the direction and apply the rotational analysis of summing the torques.  For example, if you guess a torque to be ccw(+), and you get a negative number for the value of that torque, then you define the direction incorrectly initially, if the value was positive, then you guess correctly (assuming no algebraic mistakes).
  • Scaling forces and distance on an e-FBD can help you visualize the contribution of each torque from each force.  When finished solving the problem, you can then go back to your scaled drawling to see if the answer is what you were expecting.

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways. 

Physical

Physical Representations describes the physical phenomena of the situation in a visual way.

 

We represent the system with an extended free body diagram which places the forces at locations, lets consider two kids on a seesaw. 

This is an image of two children on a see-saw. there is a boy sitting at some distance r one from the center to the left with a mass one and weight one pointing down. There is a girl sitting at some distance r two from the center to the right with a mass two and weight two pointing down. On the center of the see saw is a force that points up that is equal in magnitude to the sum of both the weights.

Naturally your not expected to draw something as complex as the above, you should draw something like the following.

This is a representation of the free body diagram of the system with the addition of a normal force pointing up for both of the masses.

Mathematical

Mathematical Representation uses equation(s) to describe and analyze the situation.

Forces acting on an object can cause torque about some given axis. Torque ($\tau$) is defined as a cross product between force and displacement.

$\tau = r \times F$,   r is the displacement vector from the axis of interest to the location the force is applied.

Remember to take geometry into account when calculating torque! Only the force component perpendicular to $r$ contributes to the torque.

$|\tau| = |r| |F| \sin \theta$

With regards to our seesaw scenario, the net torque of the see saw about its pivot axis is,

$\sum \tau = r_1 \times F^{N}_1+r_2 \times F^{N}_2$

If $\tau_{net} = 0$, then we say that the system is in rotational equilibrium. 

The net torque and moment of inertia about a chosen axis tells us something about the angular acceleration of that body about that same axis.

$\alpha = \frac{\sum \tau}{I}$

A representation with the words newton's second law on the top. There is an equation that shows that the net torque from forces external to the system acting about axis naught, is equal to the product of the moment of inertia about axis naught, and the angular acceleration about axis naught. this is also written in words below.

A representation with the words torque on the top. There is an equation that shows that the magnitude of the torque is equal to the magnitude of the position vector multiplied by the magnitude of the applied force vector multiplied by the smallest angle between the position vector and the force vector. this is also written in words below.

A representation with the words moment of inertia on the top. There is an equation that shows that the moment of inertia is found by adding up all the pieces of mass multiplied by the square of the distance from each piece of mass to the axis of rotation. This is also written in words below.

A representation with the words moment of inertia point particle on the top. There is an equation that shows that the moment of inertia of a point particle is equal to the product of the mass of the point particle and the square of the distance to the rotation axis. This is also written in words below.

A representation with the words moment of inertia point particle on the top. There is an equation that shows that the moment of inertia of a disk is equal to one half the product of the mass of the disk and the square of the distance to the rotation axis. This is also written in words below.

Graphical

Graphical Representation describes the situation through use of plots and graphs.

Rotational acceleration is proportional to the net sum of all the torque.

 

This is a representation to show that the rotational acceleration is proportional to the net sum of all the torque. There are two graphs where the first graph has the x axis as the angular acceleration proportional to the net torque over time. It shows a linear graph begins at zero and decreases and then increases. the bottom graph has the y axis as the net torque and the x axis as time. The linear graph is the same shape but starting at a positive y value.

Descriptive

Descriptive Representation describes the physical phenomena with words and annotations.

 Two children sit on a seesaw. Child 1 sits a distance of $r_1$ from the pivot. On the other side, child 2 sits a distance of $r_2$ from the pivot.

Experimental

Experimental Representation examines a physical phenomena through observations and data measurement.

 

Find a straight long flat piece of wood like the one in the figure below. Next, grab some objects to place on the piece of wood. Now,  place the piece of wood on top of a fulcrum and add arrange some objects on it so that the piece of wood balances on the fulcrum. You can vary the position of the wood and try placing the objects in a new arrangement to see where they need to go in order to find the equilibrium. 

This is an image of a long piece of wood with ticks from to eight that are equidistant to each other. The wood is on top of a fulcrum, a triangle that balances the wood. there are also three masses at different positions on top of the wood and they are still in balance.

 

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