torque and the human arm-v2

PHYS 250 Lab: Torques

Physics 250 Laboratory: Torque and the Human Arm

Score: _____ Section #:______ Name:_____________________________ Name:_____________________________ Name: _____________________________ Lab-Specific Goals:

• To analyze the relationship between forces and torques and their role in systems in equilibrium. • To analyze the forces & torques involved in the human arm.

Equipment List:

• Pasco GLX • Force probe for GLX • String with a clip attached at one end • Pasco Human Arm Model system with angle sensor • Clamp (for Human Arm Model) • 100 g mass for Human Arm Model • Ruler

Introduction and Pre-Lab Questions (Complete before coming to lab): Forces and torques play a huge role in the dynamics of the human body. For any body motion, there can be multiple pivots (joints) involved, each with several forces/torques due to muscles and the weight of the body parts. In the human back, each vertebrae is a separate pivot point, so just to stand still requires a complex balancing of forces and torque around each vertebrate. In this lab, we will focus on 2D motion of the forearm with motion restricted to just around the elbow joint. Even this extremely simplified model (no motion around the shoulder or the wrist and no twisting of the forearm, no other muscles acting) can demonstrate some interesting behavior. When we were just looking at force for linear motion, we only needed to know the magnitude and direction of the force applied to an object. When calculating torques, we also need to know where on the object the force is applied. To show this information, we draw extended free-body diagrams in which we show where on the object each force is applied. (Thus we can no longer treat an extended object as a point!)


Let’s look at the extended free-body diagram for the case where the forearm is horizontal and the biceps muscle is acting vertically. The arm is in equilibrium. 1) If the mass of the arm is 6 kg, what is FBiceps? 2) What force (include direction) does the elbow joint have to exert on the arm? 3) If you contracted your biceps muscle so that your hand moved 20 cm, how much did your biceps muscle have to contract? (Hint: this is a geometry problem not a physics problem.)

FBiceps

marm g

Elbow joint

30 cm

3 cm

Hand

20 cm

arm level

arm lifted


Activity 1. Torque in the Human Arm Experiment Setup In this part of the lab, we will use the Pasco Human Arm Model to investigate torque in human limbs. The parts of the model are identified above, along with the definitions of the angles reported by the angle sensor on the model. The string (biceps muscle) can be attached to the forearm in three locations, as shown below. In this lab, you’ll examine how attaching the ‘muscle’ at these different points affects the force required of the muscle.

biceps

elbow

forearm

triceps

outer biceps attachment

middle biceps attachment

inner biceps attachment

θforearm

θupperarm

hand

COM of arm (with no added weight)


The Set-up:

1) Start with the arm straight downwards (see figure at right). 2) Position the “hand” so that it is in line with the forearm (not

angled up or down). 3) Remove (if still attached) the 100 g mass at the “hand”. 4) Plug in the force probe in port 1. You will use this to pull on

one end of the string (attached to one of the biceps muscle attachments on the forearm), essentially acting as the biceps muscle.

5) Plug in the angle sensor (attached to the arm apparatus) to port 2.

6) Set up the GLX in Digits mode to read the force and both the angles. (Set it for 4 readings – press F2 to get this.)

7) The GLX will probably read Force and Angle 1 by default. You will need to tell it to also read Angle 2: a) Press the check-mark button, which should highlight

“Force”. b) Press the down-arrow button to highlight the blank title of

the third box. c) Press the check-mark button again, and a menu of options

should appear. d) Scroll down to “Angle 2” and press the check-mark

button again. e) The angles should both read close to zero degrees in this

position. (See the figure on the previous page for the definition of the angles.)

8) Lock the upper arm in place at or very near zero angle or have someone hold it so that it doesn’t move when you pull on the string (only the forearm should move)

9) Attach the string (using the clip) to the middle biceps muscle attachment point. (See the figure on the previous page to see the attachment points.)

10) Run the string as shown in the figure to the right. 11) Attach the left end of the string to the force probe. 12) Pull on the string and see how the forearm angle and force readings change. (The upper arm

angle should still stay close to zero degrees as long as the upper arm is locked in place.) First, determine some important distances from the pivot at the elbow:

Table 1: Distance data for the Human Arm Model

Inner biceps attachment

Middle biceps attachment

Outer biceps attachment

“Hand” (where 100 g mass attaches)


Activity 2. Effect of muscle attachment location and weight supported by hand For this part of the lab, we will be keeping the elbow bent at 90 degrees, so the forearm is horizontal (the upper arm remains vertical). You will vary two parameters to see their effect on the force required by the biceps muscle:

(1) the weight being supported (by adding or removing the 100 gram bronze mass that screws in by the “hand”),

(2) where the biceps muscle is attached Before we do the experiment, however, let’s make a prediction about our results! Here is the extended free-body diagram on the arm showing the forces acting on the forearm (other than the force due to the elbow joint, which exerts no torque around the joint so we are neglecting it here). Write the torque equilibrium condition (τnet = 0) around the elbow in terms of the forces and other information on the diagram. Solve for Fbiceps in your equation in terms of the distances involved, marm, and the 0.98 N weight.

FBiceps

marm g

Elbow joint

rarm

rbiceps

Hand rhand

0.98 N


What does your expression say about how Fbiceps depends on rbiceps? What does it predict will happen to Fbiceps required to hold the arm in equilibrium as the biceps muscle is attached closer to the elbow joint? What kind of mathematical relationship do you predict between Fbiceps and rbiceps: linear, quadratic, square root, inverse? Explain. Does your expression predict that having a 100 g (0.98 N, so we’ll just round up to 1 N) weight in the hand cause Fbiceps to increase by less than 1 N, 1 N, or more than 1 N from what it would be otherwise? Explain. Now it’s time to do the experiment! Determine the muscle force required to hold the arm steady (forearm horizontal) for these six cases.

Table 2: Force Measurements for the Human Arm Model

rbiceps [from Activity 1]

Biceps force with empty hand

Biceps force with 100 g added to hand

Inner biceps attachment



Graph both biceps forces versus rbiceps on the same sheet of graph paper on the next page. Discuss the mathematical relationship between Fbiceps and rbiceps – does it look like you predicted above?


Graph 1: Fbiceps vs rbiceps

Did adding 1 N of weight to the hand increase Fbiceps by 1 N, less than 1 N, or more than 1 N? Compare to what you predicted above.


Activity 3. Muscle Contraction You probably found that the biceps muscle has to exert a larger force than the weight of the object. That seems like a poor way of doing things – could there be a benefit that’s worth the tradeoff of increased force required of the muscle? For each of the biceps muscle attachments, move the arm through some angle Δθ. Measure how far the hand moves and how much the muscle has to contract (i.e., how far you have to pull the string) and enter that data below. (Hint: to calculate the distance the hand moved, you can use rhandΔθ, where rhand is the distance of the hand from the elbow and Δθ is in radians.) Also calculate the ratios of arm motion to muscle contraction:

Table 3: Distance Ratios for the Human Arm Model Δθ Distance hand

moved (cm) Muscle contraction

(cm) Ratio of hand

motion to muscle

contraction Inner biceps attachment



From your data above, what’s the benefit of having the biceps muscle attached close to the elbow joint? Summarize what you learned from Activities 2 & 3: What are the benefits/tradeoffs of having the biceps muscle attached close to the elbow joint? [This question is the most important question in this lab, so take the time to answer it well!]


Activity 4. Making a more accurate Human Arm Model Let’s see what happens if we assume that the biceps muscle attachment is on a line (shown as a dotted line below) connecting the elbow joint and the center of mass of the arm. This model is shown below with the arm at an angle θ (which is θforearm). FORCES/TORQUES ACTING ON FOREARM The net torque (=0) equation for this situation is:

rb Fbiceps sin(θ) – rM Mg sin(θ) = 0 Note that the θ dependence cancels and the biceps muscle force does not depend on the angle in this model for the arm. Test this prediction using the Human arm model with the 100 g mass attached to the hand. Use the inner biceps attachment. Measure Fbiceps as a function of θ from θ = 10 degrees to 145 degrees (the maximum possible with this setup) in steps of about 20 degrees. (If your arm won’t go up that high, then go up to the maximum possible angle you can.) Record your results in the table below. NOTE: Do not exceed the 50 N limit of the force sensor.

Table 4: Force versus Angle for the Human Arm Model (Inner biceps attachment) Forearm Angle θ (degrees) Biceps force (N)

Mg

Fbiceps

rb rM

θ


Is Fbiceps independent of θ as it should be if the biceps attachment is on the line between the center of mass of the arm and the elbow joint? Look closely at the human arm model (there is a close-up photo at the bottom of page 3). Is the biceps attachment on the line between the center of mass of the arm and the elbow joint? Now consider the following, more sophisticated model for the arm model at an angle θ (which is θforearm). The figure to the right is a blow-up of the area between the elbow and the biceps muscle attachment. Note that the biceps muscle is attached above the line connecting the elbow and the center of mass of the arm, making an angle φ with that line. FORCES/TORQUES ACTING ON FOREARM CLOSE-UP OF BICEPS ATTACHMENT The net torque (=0) equation for this situation is:

rb Fbiceps sin(θ+φ) – rM Mg sin(θ) = 0 (Our previous model had φ=0, for which case all the θ dependence would cancel and the biceps muscle force would not depend on the angle.) However, for this model (and your arm), φ > 0 and there is an angular dependence.

Mg

Fbiceps

rb rM

θ

rb

θ

Fbiceps

φ


Below is a graph for the theoretical value for Fbiceps as a function of θ for φ = 15°. Does this graph look something like your experimental data in Table 4? (We didn’t ask you to graph it for time’s sake, but you should be able to see whether your data would look something like this if you graphed it.) Note that near θ = 180° – φ = 165°, Fbiceps becomes very large! Let’s look more closely at that situation. From the close-up of the biceps muscle attachment below, what will happen to the torque due to the biceps muscle if θ + φ = 180°? How does that explain why a very large biceps force is required near this angle for the arm to be in equilibrium?

rb

θ

Fbiceps

φ


By finding the angle θ where the even a large biceps muscle force will not lift the arm any more, you can find φ. (Note: φ is too small for the outer attachment for you to find this way.) Pull the string directly with your hand – do not use the force sensor to pull the string.

Table 5: Angle of Biceps Attachment for the Human Arm Model Attachment φ

Inner (closest to elbow)

Middle

Note that since φ depends on the angle between the muscle attachment and the center of mass of the arm/weight system, you can change φ by changing the position of the 0.98 N weight in the hand. See what happens to φ when you put the weight as high in the hand as possible and as low as possible. (Do this for the innermost attachment.)

Table 6: Angle of Biceps Attachment for different hand positions (inner attachment) Position of hand with 0.98 N Φ

Angled upwards

Angled downwards

As you angle the hand upwards, the COM of the arm-hand system rises, decreasing φ. As you angle the hand downwards, the COM of the arm-hand system lowers, increasing φ. Were your results consistent with that idea?

So here’s one improvement to our model for the human arm: the biceps muscle attachment is not on a line connecting the elbow joint and the center of mass of the arm/hand! Now try lifting your own arm as we have been using the Human Arm Model. Do you feel the same angular dependence of the biceps force? Is there an angle at which you can’t lift the arm any higher? (Hopefully not!) So, what else is missing in our model? If you have taken an anatomy course, you can probably identify some of them, such as (in order of increasing importance): (1) The model assumes that the biceps muscle (biceps brachii) always pulls directly upwards, which isn’t quite true (it just makes the calculations a lot easier). (2) The biceps muscle is made up of two bundles of muscle that are attached at different locations on the scapula (shoulder blade). With two different origin attachments, the two bundles pull at different angles. (3) The brachialis muscle (brachialis anticus) on the upper arm is also responsible for flexing the elbow, so the biceps does not have to do the job alone.


PHYS 250 Torque in the Human Arm Lab PreLab Assignment Name _________________________________ Section __________ Let’s look at the extended free-body diagram for the case where the forearm is horizontal and the biceps muscle is acting vertically. The arm is in equilibrium. 1) If the mass of the arm is 6 kg, what is FBiceps? 2) What force (include direction) does the elbow joint have to exert on the arm? 3) If you contracted your biceps muscle so that your hand moved 20 cm, how much did your biceps muscle have to contract? (Hint: this is a geometry problem not a physics problem.)

FBiceps

marm g

Elbow joint

30 cm

3 cm

Hand

20 cm

arm level

arm lifted

torque and the human arm-v2

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