Equilibrium of Forces Acting at a Point

Transcription

1 Eqilibrim of orces Acting at a Point Eqilibrim of orces Acting at a Point Pre-lab Qestions 1. What is the definition of eqilibrim? Can an object be moving and still be in eqilibrim? Explain.. or this lab, what criteria will yo se to decide when the forces acting on the ring are in eqilibrim? 3. Draw a set of three vectors whose sm is zero. 4. How will yo estimate the ncertainty of each force acting on the ring? 5. How will yo estimate the ncertainty of each angle measrement? 6. How mch error will be introdced if the table is not level? How cold yo test this empirically? 7. What are the nits of sine and cosine? Does yor answer depend on the nits of the angle (degrees or radians)? Introdction Addition of orces orces are one of a grop of qantities known as vectors, which are distingished from reglar nmber (know as scalars) by the fact that a vector has two qantities associated with it, a magnitde and a direction (related to a coordinate axes of the system yo are dealing). These properties completely characterize a vector. A vector may alternatively be described by specifying its vector components. In the case of the Cartesian coordinate system (the system we will be primarily dealing with) there are two components, the x-component and y-component. These two properties also completely characterize a vector. Vectors, and in the case of this lab, force vectors, can be represented pictorially (see ig.1) by an arrow pointing in the direction of action of the force, with a length proportional to the strength (magnitde) of the force. The components x and y in the x and y directions of the vector are related to the magnitde and angle θ by: and conversely: x = cosθ and y = sinθ θ x igre 1 y y = x + y, and θ =arctan x. Department of Physics and Astronomy 47

2 Eqilibrim of orces Acting at a Point When several forces act on a point, their sm can be obtained according to the rles of vector algebra. Graphically, the sm of two forces = 1 + can be fond by sing the parallelogram rle illstrated in ig. a or eqivalently, by the head-to-tail method illstrated in ig. b. 1 1 Parallelogram Rle Head-to-Tail Method igre a igre b The sm of the vectors can also be derived analytically by adding their components: x = 1x + x, and y = 1y + y. Condition for Translational Eqilibrim An object is in translational eqilibrim when the vector sm of all the forces acting on it is zero. In this experiment we shall stdy the translational eqilibrim of a small ring acted on by several forces on an apparats known as a force table, see ig. 3. This apparats enables one to case the forces of gravity acting on several masses ( = mg) to be broght to bear on the small ring. These forces are adjsted ntil eqilibrim of the ring is achieved. Yo will then add the forces analytically by adding their components and graphically by drawing the vectors and determining if they add to zero sing the rles for the addition of force vectors listed above. Procedre Part 1. Eqilibrim with Three orces We shall first stdy the eqilibrim of the small ring when there are three forces acting on it. Two of the forces ( 1 and ) will be fixed and the third one 3 adjsted ntil eqilibrim is reached. 1. If necessary, level the force table sing the small bbble level placed on the table s srface.. Choose any two masses yo like in the range g, and place each mass on a weight holder. Designate the measred masses as m 1 and m, and se an electronic balance to measre each of the masses inclding the holder. The ncertainty of these measrements will be determined by either the precision or accracy of the balance, whichever is greater (see Instrment Accracy Ratings) 48 University of North Carolina

3 Eqilibrim of orces Acting at a Point 3. Place the pin in the middle of the force table and place the ring over the pin. Attach two of the for plleys provided to the force table at any position other than zero degrees. Record the vale of θ 1 and θ. The ncertainty in these angles shold be limited to the precision to which yo can read the angles on the force table. 4. Rn two of the strings (attached to the ring) over the plleys, and sspend the masses that yo have chosen at the appropriate angles (m 1 at θ 1 and m at θ ). The two strings pll on the ring with tension forces 1 and, each with a magnitde approximately eqal to the weight of the corresponding mass and holder (m 1 g, and m g) sspended at the end of each of the string. 5. Pll one of the remaining strings in varios directions ntil yo locate an angle in which the ring is freed from the pin when yo apply the right amont of force. Attach a third plley at this position. Rn the string over the plley and attach a weight holder to the string. Add weights to the weight holder ntil the ring is centered arond the pin and plls away from the pin, so that the pin is not necessary to hold the ring in place. This last added force is the (eqilibriant) force 3 (m 3 g). It may be necessary to make minor adjstments to the angle to obtain a precise measrement. Make sre that the strings are stretched radially and the pin is at the center of the ring. Gently tapping on the table will redce the effect of friction from the plleys. Estimate the ncertainty in the eqilibrant force by adjsting the mass and angle ntil the system is no longer in eqilibrim. Part. Eqilibrim with or orces 1. Now select three masses (to provide three forces with sm ) at three angles (one of them zero) and determine what forth single mass and angle establishes eqilibrim on the force table (the eqilibriant force 4 ).. Record all angles, masses and their ncertainties as in part 1. Be sre to pledge yor work, initial yor data, and have yor TA initial yor data. Analysis Part 1. Graphical Analysis Make accrate diagrams on rectanglar graph paper showing the sm of the forces acting on the ring for both parts of the experiment above (eqilibrim for 3 and 4 forces) 1. Draw force diagrams to scale. or example, 5 Newtons = 1 cm. Use whatever scale works best to give yo the greatest plotting precision.. Use the head-to-tail method to find the sm of the forces graphically. Be as accrate as possible. Qalitatively verify that the sm is zero. If it is not, determine from yor graph the magnitde of the deviation from zero. Department of Physics and Astronomy 49

4 Eqilibrim of orces Acting at a Point Part. Analytical Sm Calclate the resltant force on the ring, T = , analytically for Part 1 only. Choose zero degree to be the +x-axis, and 90 to be the +y-axis. A data analysis sheet is provided to facilitate the error analysis. 1. Use the tables on the data analysis sheet to enter the data for the forces acting on the ring. or each force, inclde the magnitde, its ncertainty, the direction θ, and its ncertainty θ. The vales of θ mst be expressed in nits of radians.. Calclate the x- and the y-components of each of the forces together with their errors. Pay attention to the sign of each component. Inclde in the last row of the table the sm of the components and their error. 3. Calclate the magnitde and the direction of the resltant force T. Compte also the ncertainty of the magnitde. Discssion Is the condition for static eqilibrim, T = 0, satisfied for both parts of the experiment? How does yor ncertainty of T compare to the precision of yor force and angle measrements? Discss the sorces of systematic error and how they affect yor reslts. What is the primary sorce of error in this experiment? Discss attempts yo have made to redce both systematic and random errors. What did yo learn or discover from this lab? When might yo apply the skills learned from this lab? 50 University of North Carolina

5 Eqilibrim of orces Acting at a Point DATA ANALYSIS Magnitde and Direction of Applied orces: orce 1 orce orce 3 m (g) m (g) (N) (N) θ (deg) θ (deg) Sample Calclation: (orce 3) Note: Newton (N) = kg*m/s = mg = = g m = Components of the Applied orces: orce 1 x (N) x (N) y (N) y (N) orce orce 3 Resltant Sample Calclation: (Components of orce 3) x = cosθ = x = y = sinθ = y = Department of Physics and Astronomy 51

6 Eqilibrim of orces Acting at a Point Resltant orce on the Ring: (N) (N) Resltant Sample Calclation: ( x, y, x, and y from Resltant row) = + = x y = Does ± = 0 (as reqired for eqilibrim)? 5 University of North Carolina