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MVS330

Hints for Problem set 5

You will need to use the text book for this problem set. If you do not have access to a copy, you can use the online google book version which you can find through Mirlyn or via this link

The Parallel axis theorem is used to find the moment of inertia at a different point of rotation.
$ I_{o} = I_{CoM} + mr^2 $
where $ m $ is equal to the mass of the object in kg and $ r $ is equal to the perpendicular distance between the CoM and new point of rotation.

The table you will use for this problem set (table 2.6) gives the moment of inertia about the CoM of the segments. The somersault axis represents the transverse axis and is the axis of interest for problem 6.

In problem 5, the coefficient of drag is given as $ 0.6m^2 $ this is not correct, the coefficient does not have units ($ C_{D}=0.6 $)

The production of work based on oxygen consumption is $ \frac{20.1KJ}{LO_{2}} $

$ GrossEfficiency = \frac{WorkOutput}{WorkInput} $

$ NetEfficiency = \frac{WorkOutput}{WorkInput - RestingMetabolic} $

I highly recommend putting in your units when solving problem 5.

In table 2.5, you are given an average segment length for men and women (in cm), but a segment weight as a percentage of the subject's body weight and the location of the CoM as a percentage of segment length.

In problem 6c, you need to find the angle between the muscle and the arm in order to calculate the moment arm of the muscle (I recommend using a diagram to find the angle).

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Instructions for handing in Problem Sets

Here are the instructions for turning in problem sets, this is important to follow not only for grading turn around time, but also because it is good practice for this type of problem solving.

Please note, you will not get full points if you do not follow these instructions - having the correct answer is insufficient, the proper steps need to be included as well.

  • Include your name and section number on the first page
    • Wednesday 2pm - 006
    • Thursday 3pm - 007
    • Thursday 4pm - 008
    • Friday 9am - 004
  • Each new question (number) must be started on a new page. The different parts of a question (letters) can continue on the same page as long as they are clearly distinguishable.
  • Each part of a question (letters) must follow all six steps. Label each step.
  • When a question contains multiple parts (ex. a, b and c),
    • in step 2 of the first part (a) you must included all variables given in the problem, as well as your drawings (free body diagram and reference frames are required, sketches are highly recommended)
    • in the subsequent parts (b, c, etc...) only the variables needed to answer the question need to be included (make sure to add the values of the previous answers).
    • if the drawings for these section (b, c, et would be the same as that from a previous section, you do not need to redraw them - if the previous drawing is insufficient, included the new drawings in step 2
  • Before adding numbers to an equation - the variable you are solving for must be isolated (alone on one side of the equation).
  • Equations must be aligned vertically (preferably with the equal signs in line).
  • Points will be taken off for any scribbles or crossed of sections (recopying your problem sets is highly recommended, not only so that they look neat, but because it will help you to understand to problem better!)

    • If you have any questions about these instructions, please contact me.

    and remember - a happy grader equals better grades :)

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Hints for Problem set 3

Be aware of your units: $ 1ft = 0.3048m $ $ 100cm^{2} = 0.01m^{2} $

Be sure to answer the questions, for example in question 1 are you asked for the force of the swimmer or the force of the water?

Line up your reference frame according to the vertical, medial-lateral and anterior-posterior axes of the human.

When dealing with a projectile motion, remember that there are forces acting in the vertical direction, but not the horizontal direction. You must separate a velocity into its vertical and horizontal components before calculating changes in velocity.

With a projectile motion, the vertical velocity of the object when it has fallen back to its initial height will be equal and opposite to the initial vertical velocity.

There are questions which are best split into parts, for example in question 3 consider the jump, the dive and the landing in water as three separate problems (each have their own initial and final velocity).
This also applies to question 4.

The graphs for question 5a) need only be qualitative – no need for calculations. However you need to include numerical values to the axes. (I would suggest doing parts b and c before drawing the graphs to have a better idea of what the movement will look like)

Flexion is considered a negative movement.

When acceleration is constant, the same equations used for linear movement can be applied to angular movement, for example $ \omega_{f} = \omega_{i} + \alpha t $

A free body diagram includes only the segment of interest and the external forces.

Kinetic Energy: $ E_{k} = \frac{1}{2}mv^{2} $
$ \Delta E_{k} = \frac{1}{2} m (v_{f}^{2} - v_{i}^{2}) $

Potential Energy: $ E_{p} = mgh $
$ \Delta E_{p} = mg(h_{f} – h_{i}) $

Total Energy: $ U = E_{k} + E_{p} $

Mechanical Power: $ P_{mech} = \frac{U}{\Delta t} $

Conservation of Mechanical Energy:
$ \Delta E_{k} + \Delta E_{p} = U_{nc} $

If there are no nonconstant forces ($ U_{nc} $) then
$ \Delta E_{k} + \Delta E_{p} = 0 $
and therefore
$ E_{k} + E_{p} = constant $

Energy and signs
When working with Energy (rather than vectors) you can use absolute values, however you must than keep track of whether your answer is positive or negative (ex. has the potential energy increased or decreased, is the speed positive or negative)

Friction
Friction is a force (N), U is energy (Nm). In order to determine the effects of friction on the total energy, you must consider the distance it was applied over.

To calculate friction, you must use the force which is perpendicular to the surface (normal force).

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FAQ

Problem set 2

Q Do you have to show your work for questions 1-26 ?
A No, you do not have to show your work, but make sure you understand why something is true/false, a/b/c/d/e. For questions 27-29 show your work in the 6 steps.

Problem set 1

Q Do you need to indicate the number of each step (ex. “step 1”)?
A Although this is not required, I recommend doing it to keep your homework as clear as possible

Q Do you have to repeat steps 1 and 2 for the second portion of a question (i.e. if you wrote them in a, do you need to write them again for b)?
A For problem set 1, repeat steps 1 and 2 for each portion of a question (we can discuss this for future problem sets)

Q For question 8, do you have to do the graph by hand, or can you do it by computer?
A You can do question 8 either by hand or computer.

Q For question 4, do you have to show the calculations if the answer is in the text book?
A *** modified answer*** see optimal angle post

Q Does the homework need to be written in ink?
A The homework does not need to be written in ink, pencil is fine. Just make sure that it is neat!

Q Is green engineering paper required for the problem sets?
A No, any type of paper (within reason!) is acceptable.

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Hints for Problem set 2

Anthropometric Table

There is a table in your text book which lists estimates of the weight and location of the center of mass of various body segments. Remember there is a difference between weight and mass. Also be sure to indicate from which joint you are giving the location of the CoM.

Angular Motion

Linear velocity $ v $ is a function of angular velocity $ \omega $ and the distance from the center of rotation $ r $

$ v = r \omega $

$ a_{tangential} $ represents the change in magnitude of $ \textbf{v} $ and $ a_{radial} $ accounts for the change in direction of $ \textbf{v} $

$ {a_{tangential} = r \alpha} {a_{radial}= r \omega^{2}} $

You can determine if an angular velocity is positive or negative using the right hand rule - curve your fingers in the direction the object is rotating, if your thumb points down the velocity is negative, if it points up it is positive.

In order to use these equations, you need to have your angle values in radians. Radians are special in the sense that if you multiply something with a unit (example meters) by a value in radians, your answer will not have radian as a unit.

$ X{m/s} \times Y{rad} = XY{m/s} $

One last hint - if you throw something from a moving car (do not actually do this!), you need to add the velocity of the car to the velocity you threw the object with in order to get the velocity of the object relative to the road. Depending on if you throw the object in the same direction as the car is moving or in the opposite direction, the velocity of the object will be faster or slower.

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Weekly Quiz answers

Week 2

  1. What is planar motion?
    • A combination of linear and angular motion in a single plane - rotation around a point that is moving
  2. During a jump, in what order are the lower limb joints activated?
    • Hip, Knee, Ankle
  3. How do you calculate total linear acceleration?
    • $ a_{Total Linear} = a_{radial} + a_{tangential} $
    • $ a_{radial} = r \omega^2 = \frac{v^2}{r} $
    • $ a_{tangential} = r \alpha = \frac{\Delta v}{\Delta t} $
  4. In a projectile motion problem, if you are given an initial velocity, what do you have to do to the vector in order to determine maximum height?
    • You resolve the vector into its vertical and horizontal components
    • $ v_y = v sin\Theta $
    • $ v_x = v cos\Theta $
  5. With regards to constant acceleration, when would you use the quadratic equation?
    • When you need to solve for time using $ \Delta r = v_i t + \frac{1}{2} at^2 $ and $ v_i \neq 0 $

Week 1

  1. What are three methods of collecting motion capture data?
    • Video and markers
    • Fluoroscopy
    • Accelerometers
  2. During one stride, how many double support phases are there?
    • Two phases of double support in one stride
  3. What is the main difference, in terms of the Center of Mass, between walking and running?
    • During walking, the CoM is highest at mid-stance and during running the CoM is lowest at mid-stance
  4. What are two equations which can be applied when acceleration is constant?
    • $ v_f = v_i + at $
    • $ r_f - r_i = v_i t + \frac{1}{2} at^2 $
  5. What is a third equation which can be applied when acceleration is constant?
    • $ v_f^2 = v_i^2 + 2a(r_f - r_i) $
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Significant Figures

rules of thumb:

the number of significant figures is equal to the number of figures in a measured variable, not including zeros at the start of the measure.

example:
0.075 has two significant figures whereas 45.50 has four significant figures

addition and subtraction:
The result has as many decimal places as the measured value with the least number of decimal places

example:
0.075 has three decimal places whereas 45.50 has two decimal places
therefore 0.075 + 45.50 = 45.58 the answer has two decimal places

multiplication and division: the result has as many significant figures as the measured value with the least number of significant figures

example:
0.075 has two significant figures whereas 45.50 has four significant figures
therefore 0.075 x 45.50 = 3.4 the answer has two significant figures.
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Optimal angle for maximum displacement

First look at the total horizontal displacement (from initial to final position)
***not only half the distance***

Since the horizontal velocity is constant you can write the total horizontal displacement as follows:

$ \varDelta r_x = v_xt $

We now want to modify this equation so that we have the total displacement at the final time. In order to do this, we take advantage of the fact that the time from the initial position to the highest point is equal to the time from the highest point to the final position:

$ t_{total} = 2t_{up} $

Also, since we know that at the highest point the vertical velocity is equal to zero, we can isolate time from the following equation:

$ v_m = v_{yi} + at_{up} $

$ t_{up} = \frac{v_m - v_{yi}}{a} $

Since vertical velocity at maximum height is equal to zero we can write

$ t_{up} = \frac{-v_{yi}}{a} $

But this is only half the time! Therefore, when we substitute this equation into the equation for horizontal displacement, we have to multiply it by 2.

therefore we end up with

$ \varDelta r_x = \frac{v_{xi} (-2v_{yi})}{a} $

In order to continue with this problem, we need to look at $ v_{xi} $ and $ v_{yi} $ in terms of $ v_i $

$ v_x = v_i cos\Theta $
$ v_y = v_i sin\Theta $

We now have

$ \varDelta r_x = \frac{v_i cos\Theta (-2v_i sin\Theta)}{a} $

$ \varDelta r_x = \frac{-v_i^2 2cos\Theta sin\Theta}{a} $

This equation is also given in your text book.

But where does that leave us?

If you remember your identities (or have looked them up)

$ 2cos\Theta sin\Theta = sin2\Theta $

We can therefore write

$ \varDelta r_x = \frac{-v_i^2 sin2\Theta}{a} $

Horizontal displacement is therefore a function of gravity (which we cannot change) initial velocity (we want the optimal angle for all magnitudes of initial velocity) and $ sin2\Theta $ (we need to find a theta which will maximize horizontal displacement!).

Now, you need to remember that $ sin() $ can vary between values of -1 to values of 1.

To answer question 4
think of what the value of $ sin2\Theta $ needs to be in order to maximize horizontal displacement and solve for $ \Theta $

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Midterm 1 practice tests explanations

Exam A

1:why is it graph A?
In this problem, Ashley starts standing (not moving therefore initial velocity = 0) and ends siting (not moving therefore final velocity = 0). If we assume that up is positive, she is moving down in order to sit and therefore her velocity is negative. Assuming she is sitting down smoothly (not dropping on the chair) she will start off moving slowly, gain speed then slow down before hitting the chair.

9: how do you find the initial velocity?
This is where you'll be happy you did the proof for the distance equation in problem set 1 question 4! In this problem we need to first solve for time using the x direction than insert what you found into the equation for y. In the end, you will have
$ v_{i} = \sqrt{\frac{-a \Delta r}{cos\theta sin\theta}} $

this equation is the distance equation rearranged
$ \Delta r = \frac{-v_{i}^2 2cos\theta sin\theta}{a} $
which you can see how to derive here

11: What was the average force to throw the ball
In this question you need to use $ F=ma $ since you know the change in velocity and the time period, you can calculate a. Just be sure to use mass and not weight in the equation!

14: coefficient of static friction so that Tom Brady does not fall
In order not to slip, the sum of the shear forces must be equal or less than the value of static friction.
The shear forces are the ground reaction forces in the x (anterior posterior) and z (lateral) directions. The value of static friction is a function of the y ground reaction force (vertical) and the coefficient of friction.

$ F_{total shear} = \sqrt{F_{gx}^2 + F_{gz}^2} $

Tom Brady will not slip if:
$ F_{total shear} \leq \mu_{s} F_{gy} $

In this problem you are given $ F_{gx} $ and $ F_{gy} $ you only need to calculate $ F_{gz} $ which will be equal and opposite to the centripetal force $ F_{c} $

$ F_{c} = m a_{r} = m \frac{v^2}{r} $

With this information you can solve for the total shear force, and then to find the coefficient of static friction:

$ \mu_{s} \geq \frac{F_{total shear}}{F_{gy}} $

Exam B

6: What is the horizontal acceleration of the horse?
As we know $ F = ma $ and since you know the horizontal force being applied to the horse by the gymnast, it is tempting to calculate the acceleration directly, however there is friction in play, and as long as the horizontal force applied by the gymnast does not overcome static friction, the horse will not move.

$ F_{static friction} = \mu_{s} F_{gy} $

in this case I am using $ F_{gy} $ as the vertical ground reaction force of the horse. Since the horse is not accelerating in the vertical direction, we know that

$ \Sigma F_{y} = m_{horse}g + F_{y_{gymnast}} + F_{gy} = 0 $

therefore

$ F_{gy} = m_{horse} g + F_{y_{gymnast}} $

This is the force you use to calculate the value of static friction.

In this case, the horizontal force applied by the gymnast is not great enough to overcome static friction and the horse will not move.

10: what are the Fe and Ff forces on the free body diagram?
The Fe and Ff are the muscle forces (extensors and flexors).

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How to do your homeworks

Your name
Question number: summary of question

Step 1: Write down which variable you need to know

  • indicate the unit
  • predict the sign

Step 2: List all known variables

  • included gravity and variables you know equal zero
  • Draw a free body diagram
  • indicate your reference frame

Step 3: Figure out which equations allow you to solve for the unknown variable from the know variables (might need more than one equation)

Step 4: Rearrange the equations to isolate the desired variables

  • Make sure the variable names are consistent

Step 5: Plug in the known variables

  • Do NOT switch steps 4 and 5
  • one exception: if a variable has a zero value

      • Step 6: Slove

      • Put answer in box
      • With units
      • correct number of significant digits
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