Mini lab report that includes:

1)Title of the Experiment,

2)Introduction/Purpose. It should be about half of page long and describe what was the purpose of the experiment,which specific physics laws and concepts the experiment was related to, and what you were expected to observe in the experiment.

3)Validated data sheets obtained in the lab, and neatly written calculations, including all equations you used.

4) Excel graphs you generated in the lab. Graphs must-have titles, axes must be labeled, units must be specified, equations of trendlines and correlation coefficients must be displayed in the graphs.

5) Conclusion should be abouthalf of a page long and should describe your experimental results and explain whether these results were in an agreement or in disagreement with theoretical predictions. Be specific and provide "some numerical values" obtained in the lab.

\fX Cut Calibri 11 -AA = =2 ab Wrap Text General [ Copy ~ Conditional Format as Cell Insert Delete Form BIU - - CA. =EE EE Merge & Center ~ $ ~ % " Formatting ~ Table * Styles ~ Format Painter Styles Cells Font Number Clipboard Alignment Fy X V fx =C18-'Disk graph'!C18 A B D E F G H K L M N O y = 0.015978x + 0.000897 Disk & Hoop Disk & Hoop R = 0.999072 Torque Angular Acceleration (Nm) (rad/s') 0.025 0.008498458 0.55698 0.012852578 0.85465 0.019159472 1.3105 0.009785606 0.62322 0.02 0.015919265 1.0709 0.015 Torque (Nm) 0.01 slope 0.013977682 kg m 2 stov of slope 0.000246017 kg mo2. 0.005 0.004878829 22 23 24 0.2 0.4 0.6 0.8 25 1.2 1.4 26 Angular Acceleration (rad/s?) 27 28 29 30 31 32 calculations Disk graph Disk & Hoop graph Ready XX Accessibility: Investigate Type here to searchM25a: Rotational Equilibrium and Rotational Dynamics Introduction: By Newton's First Law of Motion, a body is in translational equilibrium when the net force on the object is equal to zero. However, a net force of zero can still allow an object to rotate. The tendency of a force to rotate a body about an axis is called Torque. Therefore when examining a body that could rotate and again applying Newton's First Law, the body is in rotational equilibrium when the net torque is equal to zero. For translational equilibrium the body will be either not moving or moving in a straight line at constant velocity. Similarly for rotational equilibrium the body is either not rotating or is rotating with constant angular velocity. If the body is completely stationary, meaning both not moving along a line and not rotating, then it is in static equilibrium. Newton's Second Law of Motion explains what happens when the sum of the Forces andfor the sum of Torques are not equal to zero. In these cases the body experiences acceleration as a result of the net force andfor an angular acceleration as a result of the net torque. The Second Law, in rotational form, establishes the relationship between the net torque, the moment of inertia and the angular acceleration. Moment of Inertia represents the mass for rotation. It refers not only to how much mass an object has but also to how this mass is distributed with respect to the axis of rotation. The main purpose of this experiment is to examine the properties of torque, the dynamics of rotation that occur as a result of a net torque and the moment of inertia for a rigid body. First, multiple torques will be applied to a meter stick, putting it in static equilibrium. Then, the individual torques will be determined and used to verify the rst law condition for equilibrium. For the dynamic part of the experiment torque will be applied to an apparatus composed primarily of a large horizontal solid disk and a heavy thick hoop that will mount on top of the disk. The disk or disk & hoop combination will accelerate angularly due to the net torque. The data collected will be used to determine the moment of inertia for both bodies. This experimentally determined moment of inertia will be compared to the theoretical value expected. Apparatus: Figure 1 Date Modied 03/16/22 1 Part One: Rule in equilibrium Meter stick Support stand Clamp to hold the meter stick Set of masses 2 Mass hangers Unknown mass Hangers Unknown mass Assorted masses Figure 2 Part Two: Pulley Height-adjustable stand with 3 Step Pulley center mounted bearing shaft Hoop 3-step pulley Large solid disk Heavy thick hoop >Pulley with mounting rods Mass hanger, masses and string Set of Masses Disk Caliper and metric ruler Photogate with computer timing system Figure 3 N Date Modified 03/16/22Discussion: Torque is the tendency of a force to rotate a body about an axis. It depends on the magnitude of the force, the point where the force is applied relative to the axis of rotation, and the direction of the force. It is defined by the vector equation: IT =RF sin T = Torque R = radius F = force 0 = angle (between force and radius) This expression can sometimes be simplified by using the effective lever arm. The lever arm represents the distance between the force's line of action and the axis of rotation (suspension point). It is measured on a line that is perpendicular to both (Figure 4). The lever arm's equivalency is given by the following expression: L = R sin 0 Suspension point Position _ PositionR Lever Arm L Lever ArmR Mass Mass R F 1 FR Figure 4 Combining these two expressions, the magnitude of the torque can be calculated using: 171 = [FIL Where the magnitude of the applied force on this situation (Figure 4) is the force of gravity of the hanging mass: IF| = mg When examining the direction of the torque, it is usually considered positive when the force tends to produce a counterclockwise rotation about the axis, and negative when the force tends to produce a clockwise rotation. If a rigid body is acted upon by a system of torques, where the sum of these torques is zero, the rigid body is in equilibrium with respect to rotation. This means that the body can only have two rotational motion states: to be at rest or to rotate uniformly (constant angular speed) about a fixed axis. Date Modified 03/16/22 3For the cases where the sum of the torques is not zero, the body that has the net torque applied to it is going to experience an angular acceleration. The angular acceleration will be in the same direction as the net torque. This dynamical behavior of rotation has analogous components to the behavior of translational motion where torque (1') replaces force, angular acceleration (a) replaces linear acceleration and moment of inertia (I) replaces mass. Again, when the net torque is not equal to zero, the rigid body experiences an angular acceleration in the direction of the net torque. This behavior is described by Newton's second law for rotation: 1' =16: net The mass for rotation is replaced by the moment of inertia of the body. It describes not only how much matter a body is composed of, but also how that matter is distributed about the axis of rotation. [t is a geometric characteristic of the object, as it depends only on its shape, its mass distribution throughout the shape and the position of the rotation axis. In its discrete form it is dened by: N _ 2 I E mfr, 1:1 This expression means take each particle of mass multiplied by its radius squared (where the radius is the distance from the center of the mass to the center of the axis of rotation) and add them all up to get the total. I = mir"; + mzr + mgr: + 11141:: \".mwr, For a disk with radius (R) and mass (M) rotating about (2) axis, the moment of inertia can be calculated by: Figure 5 For a thick-walled cylindrical tube with open ends, inner radius (r1), outer radius (1'2) and mass (m) rotating about (2) axis, the moment of inertia can be calculated by: Date Modied 03/16/22 4 Figure 6 The angular acceleration of the rigid body can be related to the linear acceleration (may be called tangential acceleration) at some point, a radial distance out from the axis of rotation. For cases like this experiment where the system is rotating with a constant angular acceleration (a), it is possible to calculate the linear acceleration (a) for a point located at distance (R) from the center of rotation using the following formula: :1 = Ra The rotation in this experiment is produced by an external applied torque. The force for this torque is from the tension in an attached string (FT) due to a mass hanging on the other end. The weight of the mass is greater than the tension in the string and results in the mass moving with acceleration (a) downward. The magnitude of the tension can be calculated by the following: Fr =m(g_a) Date Modg'ied 03/16/22 5 For additional information on these concepts please read/review the following sections in your textbook Torque & Equilibrium: Walker. Physics. Chapter 11 section 1 & 3 Cutnell & Johnson. Physics. Chapter 9 section 1 & 2 Rotational Dynamics & Moment of Inertia: Walker. Physics. Chapter 11 section 5 Chapter 10 section 5 Cutnell & Johnson. Physics. Chapter 9 section 4 Date Modified 03/16/22Procedures: This experiment consists of two separate parts. The rst part is conducted in order to calculate torque and verify rotational equilibrium using the meter stick and the set of masses as shown in Figure 2. The second part is conducted in order to examine rotational dynamics by varying the torque applied to the disk and hoop, determining the resulting acceleration and ultimately calculating the moment of inertia. This part uses the apparatus shown in Figure 3. Part 1: Rotational Equilibrium 1. Date Modied 03/} 6/22 Record the position of the point of suspension where the meter stick is balanced with no masses attached to it. This represents the center of mass of the meter stick, and can be found by looking at the center, point of suspension, of the clamp. The position should be recorded using units of meters with precision to the millimeter in Table 1. For each trial measure the total mass placed on each side (combined mass and mass hanger) using the most precise balance available. Remember to convert to kilograms. Place a 100- gram mass and hanger on one side of the meter stick by hanging it from the string at some convenient location. Hang a SO-gram mass and hanger on the other side of the meter stick, again from the string but now moving it along the stick until the meter stick reaches equilibrium and is level to the table. Record the position for each mass, using meters, in Table 1. Calculate the lever arm for each mass. This is the distance from the mass to the point of suspension. Next calculate the force and then the torque for each side. Note: the system has reached equilibrium; therefore both values for torque should be identical, or at least very close. For trial 2 place one of the known masses, from the mass kit, on the le: side and the unknown mass provided for the right side. Note: Make sure the masses di'erfmm those used in trial I . Again position them so equilibrium is reached. Complete all the calculations for your known mass, entering the information on Table 2. For the unknown mass, record its position and lever arm. Since we are dealing with a mass of unknown quantity, the torque cannot be directly calculated. But since the system is at equilibrium, we can assume their torques to be identical. Record the same torque value obtained for the known mass as also being the unknown's torque. Use this torque to calculate the force and then the mass of the unknown. Record your results on Table 2. Part ll: Rotational Dynamics Varying the Radius - Disk 1. 7. The step pulley under the disk has three different radii. Measure each diameter and calculate the radius converting them into units of meters. Record the data in Table 3. Mount the large disk above the step pulley on the center shaft aligning the at edge of the shaft with the disk. Add 100- grams to a mass hanger and measure the total mass of the combination. The mass will remain constant during this part. Convert the mass to kilograms and record the data in Table 3. Attach the mass hanger to the string. Wind the string on the largest radius until the mass hanger is suspended close to the outside pulley. Adjust the position of external pulley so that the string is aligned with the rod of the pulley. If necessary ask a Lab Assistant to provide computer related instructions to begin the computer data collection and analyses. The computer will display a graph of angular velocity over time. Obtain the statistical slope of the line to nd the angular acceleration and its standard deviation for each trial. Make sure to collect at least five signicant digits for the angular acceleration. Note: when conducting each trial, press start to begin collecting data rst and then release the mass. Press stop when the mass hanger reaches the oor. Repeat these steps using the other two radii. Record all of your data in Table 3. Varying the Force Disk 8. 9. 10. 11. Place a "IS-gram mass on the mass hanger and measure the total mass. The radius will now remain constant during this part of the experiment using only the middle radius. Wind the string until the mass hanger is suspended close to the outside pulley adjusting its position so it is aligned with the mounting rod, as before. Obtain the angular acceleration and its standard deviation as in the above trials. Record the data in Table 3. Conduct one more trials, using 125 grams with the mass hanger. Again obtain the angular acceleration and standard deviation as before. Record your data in Table 3. Varying the Radius and the Force Disk and Hoop 12. 13. Next add the hoop on top of the disk being carful center it in the middle of the disk. Repeat the sequence of steps from 2 through 1 l as above but now with the hoop added to the top of the disk for each trial. Record all of this data in Table 4. Theoretical Moments of lnertia 1. 2. 3. Date Modg'ied 03/16/22 Measure the diameter of the disk. Measure the inside and outside diameter of the hoop. Measure the mass of the disk and the hoop separately. Analysis: Part ll: Rotational Dynamics 1. 2. 3. For the rst graph of torque vs angular acceleration use the data on Table 3. Put the torque data Calculate the linear acceleration (a), tension (FT) and torque (1:) for the rst trial in Table 3. After you have the values calculated for the rst trial have a lab instructor conrm the result. The lab instructor will guide you on the remaining calculations for Tables 3 & 4. on the yaxis and the angular acceleration data on the x-axis. This data should represent a linear relationship. The slope of the statistical linear regression line will be the experimental moment of inertia for the disk. For the second graph of torque vs angular acceleration use the data on Table 4. Put the torque data on the yaxis and the angular acceleration data on the x-axis. Again this data should represent a linear relationship. The slope of the statistical linear regression line will be the experimental moment of inertia for the disk and hoop. Calculate the experimental moment of inertia for the hoop by taking the difference between the two moments, disk and hoop minus disk. Using the dimensions and masses measured for the disk and the hoop; calculate the theoretical moments of inertia for each. Also calculate the percent error for each comparing the experimental value to the theoretical value. Date Modied 03/16/22