LVang+-+L19+-+Torque+and+Equilibrium+Lab

Lue Vang Mr. Kellogg AP Physics - Pd. 7 6 April 2012 Performed On: 4-5 April 2012 Lab Partners: Sir Sebastian Alvarez, Esq., Lt. Lue Vang

**Torque and Rotational Equilibrium **

This lab will study torque, specifically balancing two levels of torque on two sides of a pivot point to achieve rotational equilibrium. The lab will study the effects of having variable masses at variable distances from the pivot point.
 * __Purpose: __**

 **__Background:__**  The lever arm will be the distance of the placed mass away from the pivot point. The percent difference between two of the same variables of the same units is the difference between the two values divided by the average of the two values.

 Since rotation is caused by a net torque, balancing the torques on either side of the fulcrum should result in a stable, balanced meter stick. Since torque is r ⊥ F, moving the masses further from the fulcrum should increase the torque on that side of fulcrum. As it is, a light weight can be balanced with a large weight by placing the light weight closer to the edge of its side and the large weight closer to the balance point.
 * __Hypothesis: __**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **__Apparatus:__**
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Meter stick (1)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Unknown masses (4+)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Attachable hangers (4+)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Fulcrum mechanism (1)

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">**Case I: Two Known Masses:** <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">First, balance a meter stick on the fulcrum. Hang a 100g mass at 15.0cm on the meter stick and hang a 200g mass on the other side so that the meter stick is balanced on the fulcrum. Record the data and calculate torque. <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">First, balance a meter stick on the fulcrum. Hang a 100g mass at 30.0cm on the meter stick. Then, hang 200g at 70.0cm. Finally hang a 50g mass in a position that will balance the meter stick on its fulcrum. Record the data and calculate torque. <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">First, balance a meter stick on the fulcrum. Place one of the unknown masses at 10.0cm and then suspend a 200g mass on the other side so that the meter stick is balanced. Calculate a prediction for the position of the 200g and then test it. Measure and record data. <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Suspend 100g at 0.0cm of the meter stick. Slide the fulcrum position on the meter stick so that the meter stick is balanced. Record the data and calculate toqrues.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Procedure: __**
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Case II: Three Known Masses: **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Case III: Unknown Masses: **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Case IV: Meter Stick with One Mass: **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **__Data:__** <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">**Case I: Two Known Masses:** <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">lever arm = mass position – pivot point = m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> i.e.) 0.502 – 0.150 = 0.352 m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">weight = mass * gravity = mg = kg*m/s2 = N <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> i.e.) 0.097 * 9.8 = 0.951 N <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">torque = lever arm * weight = rF = m*N <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> i.e.) -0.352 * 0.951 = -0.335
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Pivot Point (m): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.502   ||   ||   ||   ||   ||
 * |||||||||| **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 1.0: Two Known Masses **  ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass (kg) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Lever Arm (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Weight Force (N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Torque (m*N) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m1 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.097   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.150   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.352   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.951   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.335   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m2 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.201   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.671   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.169   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">1.970   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.333   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">*see Table 1.0 for calculations
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Case II: Three Known Masses: **
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Pivot Point (m): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.502   ||   ||   ||   ||   ||
 * |||||||||| **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 2.0: Three Known Masses **  ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass (kg) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Lever Arm (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Weight Force (N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Torque (m*N) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m1 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.097   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.300   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.202   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.951   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.192   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m2 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.201   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.700   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.198   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">1.970   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.390   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m3 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.050   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.090   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.412   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.490   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.202   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">counter clockwise torque = τ1 + τ3 = τccw <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">i.e.) 0.192 + 0.202 = .394 m*N <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">clockwise torque = τ2 = τcw <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">i.e.) -0.390 = -0.390 m*N <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> percent difference = (difference of magnitude/average)*100 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> i.e.) 0.004 * 0.39 = 0.99%
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 2.1: Percent Difference ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Counter Clockwise Torque (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Clockwise Torque (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Difference of Magnitude(m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Average (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Percent Difference (%) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.394  ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.390   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.004   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.39   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.99   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">rotational equilibrium = 0 = τccw + τcw … … -τcw = τccw <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">-( -0.098*1.970) + ((0.502-x) * 0.490)= (0.302 * 0.951) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> *x = calculated mass position
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">A General Rule: **
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Pivot Point (m): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.502   ||   ||   ||   ||   ||
 * |||||||||| **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 2.3: Three Known Masses #2 **  ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass (kg) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Lever Arm (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Weight Force (N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Torque (m*N) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m1 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.097   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.200   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.302   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.951   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.287   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m2 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.201   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.600   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.098   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">1.970   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.193   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m3 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.050   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">?   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">?   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.490   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">?   ||
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">x = .694 m **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">percent error = |(theoretical value – actual value)/actual value|*100 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">i.e.) |(0.694 – 0.690)/.690|*100 = 0.58%
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 2.4: Percent Error ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Calculated Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Actual Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Percent Error (%) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.694  ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.690   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.580   ||


 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Case III: Unknown Masses: **
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Pivot Point (m): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.502   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Moving Mass (kg): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.201   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Weight of Moving Mass (N): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">1.970   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass of Hanger (kg): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.017   ||


 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 3.0: Position of Stationary Masses ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Stationary Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Stationary Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Lever Arm of Stationary Mass Position (m) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">B || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.100   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.402   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">D || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.100   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.402   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">E || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.100   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.402   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">F || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.100   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.402   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">R || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.100   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.402   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">*see Table 1.0 for lever arm calculations <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">*Mass B had .04kg added to be able to fit it onto the meter stick
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 3.1: Position of Moving Masses ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Stationary Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Moving Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Lever Arm of Moving Mass (m) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">B || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.969   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.467   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">D || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.983   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.481   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">E || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.946   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.444   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">F || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.929   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.427   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">R || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.801   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.299   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">calculated stationary mass = actual mass + mass of hanger = ma + mh <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= ((lever arm of moving mass * weight of moving mass)/(lever arm of stationary mass*9.8)) - mass of hanger <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= ((rm * wm)/(rs * g)) - mh <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= (-(rm * 1.97)/(0.402 * .98)) – 0.017 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">i.e.) (-(-0.467 * 1.97)/(0.402 * .98)) – 0.017 = .216
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 3.2: Calculate Mass ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Stationary Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Predicted Mass Value ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Actual Mass Value ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">B || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.257   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.257   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">D || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.224   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.271   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">E || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.205   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.206   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">F || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.197   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.197   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">R || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.133   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.132   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">*see Table 2.4 for percent error calculations
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 3.3: Percent Error ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Stationary Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Percent Error (%) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">B || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.19   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">D || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">17.52   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">E || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.47   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">F || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.24   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">R || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.39   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">R || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.39   ||


 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Case IV: Meter Stick with One Mass: **
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Pivot Point (m): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.293   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Actual Mass of Meter Stick (kg): || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.135   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">*see Table 1.0 for calculations
 * |||||||||| **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 4.0: Meter Stick with One Mass Data **  ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass # ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass (kg) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Mass Position (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Lever Arm (m) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Weight Force (N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Torque (m*N) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m1 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.097   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.000   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.293   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.951   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.279   ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">m2 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.135   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.500   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.207   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">1.323   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.274   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">*see Table 2.1 for Percent Difference Calculations
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Table 4.1: Percent Difference ** ||
 * **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Counter Clockwise Torque (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Clockwise Torque (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Difference of Magnitudes (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Average (m*N) ** || **<span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">Percent Difference (%) ** ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.279  ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">-0.274   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.005   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">0.28   ||  <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif';">1.69   ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">According to the data, the torque from a large force can be reduced by moving the force closer to the axis. In vice versa, a small force can produce a larger torque by applying that force farther away from the axis.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Analysis: __**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Note though, that the meter stick and hanging masses are suspended above the table; thus, there is an upward force equal to all the downward force from gravity in order for the objects to remain elevated. These “support forces” are not included in torque calculations because they cancel out and have a net force of zero, thus no net torque.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">All in all, if an object is balanced at rotational equilibrium, it is even possible to calculate the object’s mass if the clockwise and counterclockwise torques can be calculated. For example, if a uniform meter stick is at equilibrium and has a 220g suspended at 5.0cm, 120g at 90.0cm, and the support stand at 40cm, it can be concluded that ((.4m–.05m)*.2kg*9.8m/s2)=((.9m–.4m)*.12kg*9.8m/s2) + ((.5m-.4m)*MASS*9.8m/s2). Solving for mass, we get that the MASS of the meter stick is 0.1kg.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> In conclusion, this lab shows how torque can be balanced by manipulating the position of a force to or away from the axis. As predicted in the hypothesis, moving a force closer to the axis decreases torque and moving a force further from the axis increases torque. Table 1.0 shows an example of this as a small mass of 0.097kg can be balanced with a large mass of 0.201kg simply by placing the small mass further from the fulcrum and the large mass closer to the fulcrum. When the clockwise and counterclockwise torques are equal in magnitude, but opposite in direction, a net torque of zero will result in a balanced meter stick. This concept of rotational equilibrium applies to all systems of rigid bodies that rotate around a fixed axis.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Conclusion: __**