Torque OBJECTIVE To verify the rotational an translational conitions for equilibrium. To etermine the center of ravity of a rii boy (meter stick). To apply the torque concept to the etermination of an unknown mass. INTRODUCTION When an unbalance force acts on a boy, the boy has the tenency to rotate (about some axis), translate, or both. For a boy to be in equilibrium, it must have both rotational equilibrium (sum of all the torques is equal to zero) an translational equilibrium (sum of all linear forces is equal to zero). The total equilibrium conition is calle static equilibrium. In this laboratory, a meter stick will serve as a rii boy to which forces (masses uner the influence of ravity) act. From these applie forces an the torque, the concept of center of ravity an center of mass will be investiate. APPARATUS () meter stick, () meter stick support stan, () balance, (4) strin loops, () knifeee clamp, () hooke mass set, an () unknown mass with hook. THEORY Torque: When a force acts on a rii boy, that is allowe to pivot about some axis, the boy will have the tenency to rotate. The tenency towar rotation is calle torque, τ. The torque is a result of a force F that is applie perpenicular to a istance that is extenin from the axis of rotation. The equation is thus iven as: = F Equation Where, τ [N m] is the torque, F [N] is the perpenicular applie force, an [m] is the perpenicular istance. Note, both an F are vectors as well as the torque itself. Torque - Pae
To fully unerstan the concept of the perpenicular istance, let us illustrate an applie force that is present on a rii boy. Pivot Point Rii Object Fx Fiure Fy F In Fiure 8-, is the istance from the pivot point to the force F. As mentione previously, only the applie force that is perpenicular to the istance applies the torque (rotation). Thus, base on the fiure, only the y-component of the applie force contributes to the rotation of the boy. In equation form this is expresse as: = ( F sin ) Equation Which is the same form as Equation () if θ = 90 o. This will enerally be the case for torques. That is, the applie force will automatically be perpenicular to the istance. This force component can either cause the rotation, relative to the axis of rotation, to be in a clockwise (CW) or counterclockwise (CCW) irection. As only a rotational motion is establishe (non-linear), the vector irection for the rotation is set so that CW motion is esinate as the neative irection an CCW motion is esinate as the positive irection. Equilibrium: The conitions for static equilibrium, as state previously, are that the vector sum of the forces must equal zero an that the vector sum of the torques must equal zero. F =0 an Equation =0 The vector sum of the forces is equal to zero is concerne with the translational (linear) equilibrium of the rii boy. That is, the object is not movin in any of the possible linear irections, whether with a linear acceleration or with constant linear velocity. As the laboratory will make use of a meter stick as the rii boy, it is restricte from linear motion, an thus, this conition is automatically satisfie. Even if an object is not movin in a linear irection, it can still be "movin", i.e. have a rotation. The vector sum of the torques equal to zero is concerne with the rotational equilibrium of the rii boy. That is, the object is not movin in either a CW or CCW irection, whether with an anular acceleration or with constant anular velocity. Torque - Pae
It is this vector sum of the torques that will be the concentration factor in this laboratory. More precisely, the sum of the torques in the CW irection shoul equal the sum of the torques in the CCW irection, if the object is to be in rotational equilibrium. CW - CCW=0 Equation 4 Let us refer to Fiure, below, to evelop an an unerstanin of Equation 4. m m m Fiure Pivot Point Where, m, m an m are the respective masses that are applyin the force to the rii boy (ue to the influence of ravity) an,, an are the istances from the pivot point to their respective applie forces (masses); these istances are calle lever arm istances. Note, mass m provies a CCW torque to the object an masses m an m provie the CW torque to the object. Thus, in the sum of the torques, the applie force ue to m an the applie force ue to m act in the same rotational irection an therefore a. The respective torques, on the object, follows: = F = F = F Applyin Equation 5 to Equation 4 yiels: Equation 5 m F m + = or + F or + m + m or Equation 6 Torque - Pae = F
Note, in the last line of Equation 6, the acceleration ue to ravity was cancele out as it appears in all terms an is, therefore, unnecessary for the computation. This form of the torque equation will be use as the basis for future calculations in this laboratory. Note, the absence of the acceleration ue to ravity in Equation 6 oes allow for the computation of unknown masses an/or istances but oes not result in the torque of the system. The torque has units of Newton-meters which o not result from the prouct of mass times istance. The Center of Gravity: The center of ravity of an object is efine by the point on which the sum of the ravitational torques, ue to the "iniviual" mass particles in the object, is equal to zero. By "iniviual" mass particles, we mean that a rii object is consiere to be mae up of tiny bits of mass. The combination of each of these masses makes up an escribes the rii boy. For a uniform rii boy (same ensity throuhout), each particle contributes to the torque about a pivot point. Aain, each of these torques is ue to the mass particles actin uner the influence of ravity; thus, the applie force is actually weiht. The iea is to locate the point on the object where the total weiht of the object is concentrate, an thus the location where the effect on the rotation of the object is the same as that of the iniviual particles. The object is in equilibrium when it is supporte by a force equal to its weiht, an the sum of the ravitational forces actin on the iniviual masses about the center of ravity equal zero. In a symmetric object (like a meter stick) the center of ravity is locate on the symmetry axis of the object. Given a meter stick, the center of ravity is locate at the 50.0 cm mark. Supportin the meter stick here results in no rotation of the object. However, venture from this mark an the meter stick will rotate as the sum of the torques CW oes not equal the sum of the torques CCW. Gravitational forces o act on the rii boy as a whole. Thus, the location (for a uniform rii boy) of the weiht concentration is also the location of the mass concentration in the object. Thus, we can refer to the center of ravity as the center of mass as lon as the acceleration ue to ravity is constant throuhout the object. Torque - Pae 4
EXPERIMENTAL PROCEDURE a) Determine the mass of the meter stick (m). Recor this value in your ata table. b) Slie the knife-ee clamp on the meter stick an clamp it near the center of the stick. Place the meter stick-haner combination on the support stan. Ajust the meter stick from sie-to-sie, throuh the haner, until the meter stick hans level (in equilibrium). Tihten the clamp to the meter stick at this point an recor this clampe position as x in your ata table. This is the location of the center of ravity of your meter stick. Recor this value in your ata table. A iaram of EACH of the conitions stuie is to be complete on the provie sketch sheet. You are to label all parts of the sketch, incluin the associate values of each mass, position, an lever arm istance. Fiure is a sample representation of what your sketch shoul look like. o Each m an shoul be the actual numerical value associate with your ata. Conition : Two Known Masses: a) With the meter stick pivote at x, place a 00 mass at the 0 cm position on the meter stick (0 cm from zero). b) Place a 00 mass on the opposite en of the meter stick at a position that causes the system to be in static equilibrium. c) Recor the masses an positions in the ata table. Complete the require sketch for this case before movin on. Conition : Three Known Masses: a) With the meter stick pivote at x, place a 00 mass at the 0 cm mark an a 00 mass at the 68 cm mark on the meter stick. b) Place a 50 mass at a position in the system that causes the system to be in static equilibrium. c) Recor the masses an their respective positions in the ata table. Complete the require sketch for this case before movin on. Conition : Meter Stick Mass: a) Place a 00 mass at the 0 cm mark on the meter stick. b) Loosen the clamp that hols the meter stick at x. Slie the meter stick in the clamp until you are able to place the system in static equilibrium. c) Recor the mass, its position, an the location of the clamp that provie the static equilibrium position. Complete the require sketch for this case before movin on. Do not foret the m of the meter stick in your rawins an calculations when the PP is not x. o The m of the meter stick is ALWAYS at the location x no matter the conition stuie see Question # at the en of the lab! Torque - Pae 5
Conition 4: Meter Stick Center of Gravity: a) Place the clamp, on the meter stick, at the 40 cm mark. b) Place a 50 mass between 5-5 cm, a 00 mass at 0 cm, an a 00 mass at 90 cm. c) Place a secon 00 mass at a position in the system that causes the system to be in static equilibrium. ) Recor the masses an their respective positions in the ata table. Complete the require sketch for this case before movin on. Do not foret the m of the meter stick when the PP is not x. Conition 5: Unknown Mass: a) By the use of any of TWO unique proceures (PP = x & PP x), etermine the mass of the unknown provie. The use of the balance is still prohibite! b) Recor all necessary values (masses, positions, an the pivot point) in the ata table. Be sure to recor the ientification number/letter of your unknown in the ata table. Complete the require sketch for this case before movin on. c) Repeat the above instructions for a ifferent experimental conition. COVER PAGE REPORT ITEMS (To be submitte an staple in the orer inicate below) (-5 points if this is not one properly) Lab Title Each lab roup member s first an last name printe clearly Color Group Date DATA (worth up to 0 points) Data tables available as a ownloaable Excel file DATA ANALYSIS (worth up to 0 points) Require sample calculations, to be shown on a separate sheet of paper in your laboratory report, are hihlihte in yellow on the ownloaable Excel ata table spreasheet Torque - Pae 6
Useful Calculation Information To Know: The istances to be use for torque calculations are calle lever arm istances. As previously state, this is the istance from the pivot point to each force (hanin mass * ravity). This value as a positive number (a manitue) as you want to have the actual istance between the two points. o When the pivot point is not at the center of ravity (x), then the mass of the meter stick must be accounte for in the torque. As such, the lever arm istance for the mass of the meter stick is foun by subtractin the actual pivot point from x. Percentae Difference: In certain experimental settins where a value is not a known constant, a comparison may still be useful to evaluate the effectiveness of the experiment base on the comparison of two experimental results. This will not tell you anythin as to the accuracy of the experiment but will ive you an inication of how repeatable (precise) the experiment is. Below is the formula for achievin a percentae ifference comparison: = E - E E + E * 00% Where, E = is experimental result number (the CW torque), E = is experimental result number (the CCW torque), an the parallel lines in the numerator inicate an absolute value. GRAPHS / Sketches (worth up to 0 points) You shoul have a total of 6 sketches: Conition : Two Known Masses ( sketch) Conition : Three Known Masses ( sketch) Conition : Meter Stick Mass ( sketch) Conition 4: Meter Stick Center of Gravity ( sketch) Conition 5: Unknown Mass ( sketches...one for each experimental conition) GRAPH ANALYSIS (worth up to 0 points) None CONCLUSION (worth up to 0 points). What was the lab esine to show?. What were your results?. How o the results support (or not support) what the lab was suppose to show? 4. What are some reasons that the results were not perfect? Torque - Pae 7
POST-LABORATORY QUESTIONS (worth up to 0 points) ) In many cases x, the center of ravity of the meter stick, is not locate at the 50.0 cm mark. What woul cause the position of x to not be locate at 50.0 cm? ) For Conition, explain how Equation is satisfie. ) Compare to Conitions &, Conitions, 4, & 5 ha the pivot point of the meter stick locate at a position other than its center of ravity. Why is the mass of the meter stick inclue in the calculations when the pivot point is not locate at x? Why is the mass of the meter stick not inclue in the calculations when the pivot point is locate at x? Torque - Pae 8