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Physiotherapy and the elbow

Example 1 (as submitted) :

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A patient recovering from elbow surgery is attending physio to strengthen the biceps muscle. It is important that the elbow joint does not experience a force greater than 200N during the early stages of recovery. The physio chooses a 45.0 N weight (4.6 kg) for the patient to hold. If the forearm which weighs W = 22.0 N is maintained in a horizontal position by the biceps muscle supplying 348 N of force, will the force at the elbow exceed the allowable 200 N?

One way to solve this problem is to do vector summation with the cosine law to see what the resultant force is. Since the arm is being maintained in a static position, the force felt at the elbow will be equal and opposite to the resultant force of W, L and F. Normally we would solve this problem using Newton’s laws where the sum of the forces in the x direction must equal zero and the same in the y direction but this problem can easily be done using the cosine law. We could vary the Load (L) or the allowable force at the elbow such that sometimes the weight held exceeds the allowable and sometimes not. L is easy to vary by just selecting some common weights from a gym (say 1 to 5 kg). F would need to be generated from an algorithm which is given by:

I will go to the gym over the next couple of weeks and take a photo of my daughter doing the exercise if you want a photo to include.

Data Ranges: Pick a height and weight for a patient and you can calculate the forearm length and weight using the information below:

Physiotherapy and the elbow

Example 1 solution :

The following is a screen shot of the finished question for Biomechanics

See the submitted problem

A patient recovering from elbow surgery is attending physio to strengthen the biceps muscle in his arm.  It is important that the elbow joint does not experience a force greater than 161 N during the early stages of recovery.  The physio chooses a 3 kg weight for the patient to hold.  The forearm which weighs 1.5 kg is maintained in a horizontal position by the biceps muscle supplying 227 N of force.  Solve the force triangle to find the magnitude of the reaction force in the elbow?

Recal that the acceleration due to gravity, g, is 9.81

Structural Engineering: Moment of inertia

Example 2 (as submitted) :

See the finished question

Topic: Algebra

Subject: Structural Engineering

Intro: The moment of inertia of a beam is a measure of the “stiffness” of the beam. The larger the moment of inertia, the less flexible the beam is. For the same cross sectional area, two beams will deflect differently depending on if they are put on edge or laid flat.

For an I-beam, the moment of inertial about its vertical axis is given by:

Question: If I_y=246×10⁶ mm⁴, b = 295 mm, H =535 mm and t = 79 mm, determine the length of h.

Variables: I_y from 200-800×10⁶ mm⁴, b=200-500 mm, H = 500-1000 mm, t= 60-100 mm

Solution:

Example  2 finished :

The following is a screen shot of the finished question for Structural Engineering

See the submitted problem

Marine engineering marine ramp design using trigonometry

Example 3 (as submitted) :

See the finished question

Application: Marina Ramp Design
Math topic: Right Angle Trigonometry

Problem Introduction: At a Vancouver marina, a pedestrian ramp leads to a floating dock which moves up and down with the tide. The upper end of the ramp is hinged (it can rotate) and the lower end is supported by a roller that can roll back and forth along a track.

Problem ( {  } = realistic range of values for variables):
Since users of the marina carry equipment up and down the ramp, its slope must not exceed 40% {30% .. 45%}. The difference in elevation between the lowest low tide and the highest high tide is 4.0 m {3.6 m .. 5.2 m}.

1. Assuming that the ramp is horizontal when the water level is at its highest, determine the minimum length of the ramp (in m) such that its slope never exceeds 40% {as above}. Express your answer to three significant figures.
2. Using the minimum ramp length calculated above, determine the minimum length of the roller track (in mm) required to support the lower end of the ramp.

Answer Key:
Please provide a brief solution for a particular set of values given in the question provided. If applicable, indicate the range that the answer could reasonably take on (for example, if it is a physically meaningful result, certain answers may not make sense).

For the values given in the question above:

First step: find angle between the horizontal and the ramp:

1. Minimum ramp length
2. Minimum roller track length
   = Ramp length when horizontal – Horizontal distance covered by ramp when sloped
   

Example 3 finished:

The following are screen captures of the finished question for Marine Engineering.

 See the submitted problem

At a Vancouver marina, a pedestrian ramp leads to a floating dock which moves up and down with the tide. The upper end of the ramp is hinged (it can rotate) and the lower end is supported by a roller that can roll back and forth along a track. As the tide rises and falls, the slope of the ramp changes, the slope of the ramp changes, and the roller on the lower end moves in its track. Note: the diagram below is not to scale.

Since users of the marina carry equipment up and down the ramp, the slope of the ramp must not exceed 34%. The slope gives the relationship between the vertical and horizontal distances covered by the ramp (i.e. that the vertical drop can be at most 34% of the horizontal distance travelled). The difference in elevation between the lowest low tide and the highest high tide at this location is 4.1m.

Part A:

Assuming that the ramp is horizontal when the water level is at its highest *as shown above, determine the minimum length of the ramp such that its slope will never exceed 34%. Give your answer in meters, to 2 decimals

Part B:

Using the minimum ramp length calculated above, determine the minimum length of the roller track (in mm) required to support the lower end of the ramp. Give your answer in millimeters, to 3 significant figures.