Introduction This lab involves the analysis of objects that are in static equili
ID: 1461975 • Letter: I
Question
Introduction
This lab involves the analysis of objects that are in static equilibrium. An object is in static equilibrium if the sum of the forces that other objects exert on it is zero and the sum of the torques due to these forces is also zero. Static equilibrium allows us to analyze and understand body levers and many devices of everyday life.
Torque
Definition of torque: The torque caused by the force that one object exerts on another is a measure of the turning effect due to that force. Torque is defined relative to a particular axis of rotation and is found using
t = ± F L sin q
where F is the magnitude of the force and L is the distance to the force. The axis of rotation is not necessarily a physical point of rotation. It is the origin from which the distance, L, is measured. q is the angle between the direction of the force and the line along which L is measured. The torque due to the force is positive if it tends to turn the rigid body counter-clockwise about the origin and negative if it tends to turn the body clockwise. The metric unit for the torque is N•m. (The English system unit is ft•lb).
We can rewrite the expression for torque as (F sin q) L. Note in the figure above that Fsinq is the component of the force perpendicular to the beam. From this we see that the torque is just the perpendicular component of the force times the distance from the origin to the point of application.
Net torque: When multiple forces act on an object, the net torque about the origin is the sum of the individual torques:
tnet = t1+ t2 + t3 + …
where each of the individual torques is due to one of the forces, and each has a sign according to the direction in which it acts.
Center of gravity: The center of gravity of a body is defined as that point through which the torque due to the mass of the body is zero. In other words, it is the support point at which the mass is balanced. Effectively, the entire mass of the body can be assumed to act at the center of gravity. For uniform, symmetric bodies this is also the center of symmetry. For example, a meter stick that is uniform and symmetric has its center of gravity at the 0.500-m mark. Any real meter stick will probably be very close to uniform and symmetric, so its center of gravity would probably be very close to the 0.500-m mark. We can assume when finding the torque of the meter stick that its weight acts at the center of gravity.
Equilibrium
An object is in static equilibrium if it satisfies two conditions, called the conditions of equilibrium.
First condition of equilibrium
An extended body is in translational equilibrium (either at rest or moving with constant velocity) when the sum of all forces exerted on this extended body is zero:
= + + … + = 0
The subscript n is a label for the external forces exerted on the extended body. If we resolve the forces into components along any two coordinate axes, then the above condition can be rewritten as:
Fx = (F1 on B)x + (F2 on B)x + … + (Fn on B)x = 0
Fy = (F1 on B)y + (F2 on B)y + … + (Fn on B)y = 0
In words, the sum of the x components of all forces on the object must add to zero. This ensures that the object will not accelerate in the x direction. Similarly, the sum of the y components of all forces on the object must add to zero. This ensures that the object will not accelerate in the y direction.
Second condition of equilibrium
For an object to be in static equilibrium and not have a rotational or turning acceleration, the sum of all torques due to the forces on the object must be zero.
t = t net = t 1 + t2 + … + tn = 0.
This condition ensures that the object does not have an angular acceleration – that it does not start turning or change the rate of its rotation. Note that for an object in static equilibrium, the second condition of equilibrium (t net = 0) is independent of the choice of origin. That is, the sum of the torques will be zero no matter where we choose to place the origin.
Objectives
Measure the net force on a system in static equilibrium
Determine the individual and net torques on a system in static equilibrium
Use the conditions of static equilibrium to determine the magnitude of a force on an object
Use a model of the forearm to explore the relationship between the weight of a supported object and the force exerted by the bicep muscle
Materials
Computer
Vernier Computer interface
Logger Pro
Vernier Force Sensor
Ring stand
Cross bar with clamp
2 mass hangers and set of masses
unknown mass
“forearm” stick
rod that does thought the forearm and clamp
string
Procedure
Note: Throughout this lab, use string to hang the stick and weights.
Part A: The conditions of equilibrium
1)Connect the Dual-Range Force Sensor to Channel 1 of the interface. Set the range switch on the Force Sensor to 10 N.
2)Open Logger Pro. If you get a message asking about the Force Sensor, select the option to use the setting on the sensor.
3)Hang the Force Sensor from the ring stand by placing a rod through the hole in the sensor and securing it to the stand with a clamp. Tighten the lock-screw to secure the sensor. Click to set the Force Sensor to zero.
4)Hang the meter stick from the Force Sensor. Adjust the meter stick so that it is balanced and as horizontal as practical. Logger Pro should now display the weight of the meter stick. Record it on the diagram to the right along with the position of the center of gravity (CoG)
5)Place a mass hanger and additional mass so that a total mass of 200 g hangs from the left side of the meter stick, at the 15 cm mark as shown. Place another hanger with a total mass of 500g on the right side, and adjust its position so that the system is in equilibrium – the meter stick is balanced and horizontal and has zero rotational and vertical acceleration.
6)Below, record the position of the 500g block and the force displayed on Force Sensor:
Position of 500-g mass: Sensor reading:
Examine the extended body diagram below. Identify the force that corresponds to the sensor reading and label it with the appropriate value.
7)An extended-body diagram for the meter stick is shown at the right. Notice that the origin of the coordinate system is chosen to be at the center of gravity of the stick. Complete the diagram by writing in the distances to each mass from the origin, that is, from the center of gravity of the stick.
In the space below, list each of the forces acting on the stick including the correct signs, and use them to find the net force on the stick. Show your work for any calculations you do.
force due to the string holding the 200 g mass = _____________________________
force due to the string holding up the meter stick =____________________________
force due to the weight of the meter stick =__________________________________
force due to the string holding the 500 g mass =______________________________
Net force = ______________________________________________________________
Calculate the torques exerted on the meter stick by each force about the center of gravity. Record the results below. Show your work for any calculations you do. Watch your signs.
torque due to the string holding the 200 g mass = ______________________________
torque due to the string holding up the meter stick =_____________________________
torque due to the weight of the meter stick =___________________________________
torque due to the string holding the 500 g mass =_______________________________
Net torque: net =__________________________________________________________
Answer the first two Analysis questions
Part B. Determining an unknown mass using torques
In this section you will build a balance using the same principles as those used to design a scientific lab balance. You will use the balance to measure an unknown mass. Do not measure the unknown mass except using your balance.
B1)the Force Sensor from the ring stand and hang the meter stick directly from the stand so that it is balanced and level.
B2)Hang the unknown mass at the 25 cm mark on left side of the stick.
B3)Place a mass hanger on the right side of the stick. Add mass and adjust the position so that the stick is again balanced and level.
Record the position of the mass hanger and total mass below:
Position: Mass:
Answer the next two Analysis questions.
Part C. Biceps muscle system
In this section you will build a model of the forearm and use it to determine the force exerted by the “bicep” when holding up a weight.
Hang the Force Sensor from the ring stand with nothing hanging from the Force Sensor. Click to set the Force Sensor to zero.
Now hang the “forearm” stick (smaller stick with hole) from the Force Sensor. Find its center of gravity by adjusting the support point until the stick is balanced. Record the position of the center of gravity and the weight of the stick in newtons (N) as given by the sensor. Note that the center of gravity may not be at the center of the stick. (Why?)
Wstick (N): Position CoG:
A model of the biceps muscle system of the arm is shown at the right. The arm pivots at the elbow and is pulled up by the biceps. Construct an experimental apparatus that is similar to this model. Clamp the small rod to the ring stand to serve as the elbow. The stick will serve as the forearm. The Force Sensor will serve as the biceps muscle. A 175 g mass will serve as the bar bell.
Let the biceps support the forearm about 10 cm from the elbow. Hang the bar bell near the end of the forearm.
Use a cross bar to hold the top of the Force Sensor. Slide the sensor so it’s vertical, and adjust the cross bar up or down so that the forearm is horizontal. Record the reading on the Force Sensor biceps muscle in newtons (N).
An extended-body diagram for the forearm is shown at the right. Label the diagram with the distances from the elbow and known magnitudes of the forces exerted on the forearm.
Because the arm is in equilibrium, the vertical components of all the forces on the meter stick should add to zero. Use this to determine the magnitude of the vertical force of the upper arm on the forearm.
Calculate the torques about the elbow due to each of the forces. Show your calculations. Watch your signs.
torque due to the force of the upper arm =_____________________________________
torque due to the force of the biceps muscle =__________________________________
torque due to the weight of the forearm =______________________________________
torque due to the weight of the bar bell =______________________________________
Net torque: net =____________________________________________________________
Answer the remaining analysis questions.
Analysis
Part A
1.What was the net force on the stick? What was the net torque? Are your numbers what you expected? Explain why or why not, commenting as necessary on the accuracy and precision of your equipment.
2.Repeat your calculation of net torque, but this time use the zero mark on the meter stick as the origin. Clearly show the new values of each torque, and use them to determine the net torque. (Recall that we can calculate the torque about any point, independently of how an object may actually rotate.) What do you find for the net torque? Is this what you would expect? Explain why or why not.
Part B
3.similar to the diagram in Part A. This should include the distance to each force and the magnitudes of the known forces.
4.Write an equation for the net torque on the stick about the center of gravity, being sure to include all the forces on the stick. Use the fact that the stick is in equilibrium to solve for the unknown mass. Report your solution along with the number of your unknown.
Part C
5.What was the net torque about the elbow? Why didn’t you need the force of the upper arm on the forearm to find it? What would you expect to find if you calculated the net torque about the center of gravity of the forearm instead of about the elbow?
6.Compare the force exerted by the biceps muscle with the weight of the bar bell. Why is one so much larger than the other? (Hint: Think about the torque about the elbow.) What does this imply about the force exerted by your biceps muscle in relation to the weight you are able to lift with your forearm?
I would like to know how to do part A, B, and C from the Analysis
Explanation / Answer
I can give some examples like:
How can the torque applied to the wrench in the figure be expressed in terms of r, |F|, and the angle between these two vectors? . The force vector and its F component form the hypotenuse and one leg of a right triangle, and the interior angle opposite to F equals . The absolute value of F can thus be expressed as F = |F| sin , leading to || = r|F| sin . Sometimes torque can be more neatly visualized in terms of the quantity r shown in figure r, which gives us a third way of expressing the relationship between torque and force: | | = r|F|. Of course you would not want to go and memorize all three equations for torque. Starting from any one of them you could easily derive the other two using trigonometry. Familiarizing yourself with them can however clue you in to easier avenues of attack on certain
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