Static equilibrium lab report introduction

Writing Static Equilibrium Lab Report

Static Equilibrium and Rotational Balance

Introduction

In Part I of this lab, you will observe static equilibrium for a meter stick suspended horizontally. In Part II, you will observe the rotational balance of a cylinder on an incline. You will vary the mass hanging from the side of the cylinder for different angles.

Reference

Young and Freedman, University Physics, 12th Edition: Chapter 11, section 3

Theory

Part I: When forces act on an extended body, rotations about axes on the body can result as well as translational motion from unbalanced forces. Static equilibrium occurs when the net force and the net torque are both equal to zero. We will examine a special case where forces are only acting in the vertical direction and can therefore be summed simply without breaking them into components:

(1)
Torques may be calculated about the axis of your choosing:

(2)
where torque is specified by the equation:

(3)
where d is the lever arm (or moment arm) for the force. The lever arm is the perpendicular distance from the line of force to the axis about which you are calculating the torque.

Normally, up is "+" and down is "-" for forces. For torques, it is convenient to define clockwise as "-" and counterclockwise as "+". Whatever you decide to do, be consistent with your signs and make sure you understand what a "+" or "-" value for your force or torque means directionally.

Part II: Any round object when placed on an incline has tendency of rotating towards the bottom of an incline. If the downward force that causes the object to accelerate down the slope is canceled by another force, the object will remain stationary on the incline. Figure 1 shows the configuration of the setup. In order to have the rubber cylinder in static equilibrium we should satisfy the following conditions:

(4)

Figure 1: Experimental setup for Part II

The condition that the net force along the x-axis (which is conveniently taken along the incline) must be zero yields the relationship. (Prove this!)
Without static friction the cylinder would slide down the incline; the presence of friction causes a torque in clockwise (negative) direction. In order to have static equilibrium we need to balance that torque with a torque in counterclockwise direction. This is achieved by hanging the appropriate mass m.

Applying the last condition to the center of the cylinder will result in:

where r, the radius of the small cylinder (PVC fitting), is the moment arm for the mass m and R, the radius of the rubber cylinder, is the moment arm for the frictional force which accounts for M and m. Combining this equation with the equation for Ffr from above will result in:

(5)
(6)
By adjusting the mass m, we can observe how the equilibrium can be achieved.

Procedure

Part I: Static Equilibrium

Figure 2: Diagram of Torque Experiment Setup

1. Weigh the meter stick you use, including the metal hangers.

2. Attach the force sensor cords to the Interface box as you have done in previous labs. Set up the hardware in Pasco by adding two force sensors for channels A and B.

3. For today's lab you do NOT need to graph the force sensors over time, instead, drag the Digits icon in the "Displays" area. This will give you a digital readout of the force sensor data at the given time. Create two Digit displays, one for each force sensor on the meter stick. Choose “Force Channel A (N)” and “Force Channel B (N)” in the two Digit displays.

4. Remove the meter stick from force sensor hooks, click on Record button, and tare each force sensor without weight to establish zero. Then hang a known mass on each force sensor to verify that it is reading correctly. Measure the mass on the scale first. If you need to increase the precision of the display, hover the mouse over the increase digits icon shown in the figure below. This will increase the number of significant figures.

5. Keep Record button on. Set up the meter stick and force sensors as shown in Figure 2. The meter stick will be suspended from a beam via the two force sensors. (These will also be used to determine the upward vertical forces at these positions.) The force sensors must be attached vertically anywhere that yields equilibrium using the metal hangers.

6. Attach three masses (100g, 200g and 500g according to the table in step 9) to the meter stick using metal hangers. Neglect the masses of the metal hangers in your torque calculations.

7. Balance your system by moving the three weights and watching the Digits displays. All forces must be vertical to avoid difficulties, so make sure the meter-stick is level and be sure the force sensors are pulling straight upward. Adjust the location of the masses so that the force meters read almost identical forces.

8. Record the position and mass on the meter-stick for each mass.

9. Using the following data sheet to record the results, calculate the sum of the masses responsible for the positive forces and the sum of those responsible for the negative forces. (Forces 5 and 6 are the force meters.) Check to see if the sums are equal.

Mass(kg)

Force #

Force (N)

Position (m)

100 = 0.1
F1

980 = .98
0.149

200 = 0.2
F2

1960 = 1.96
0.211

500 = 0.5
F3

4900 = 4.9
0.685

143.04 = .14304
F4

1401.792 = 1.401
0.5

F5

-4.60

0.091

F6

-4.50

0.944

10. Using the zero position (x = 0 m) of the meter stick as the axis of rotation and counterclockwise torques as positive, determine the sum of the torques acting in both directions and record them on the data sheet. Check for equality between positive and negative sums within the calculated uncertainties. Now imagine the lever arm is located at the axis point in the middle of the meter stick (x = 0.50 m) and recalculate torques. Check for equality between positive and negative results, within calculated uncertainties.

Torque Calculation Table

Lever Arm (m)

Axis at x=0m

Torque (N-m)

Axis at x = 0m

Lever Arm (m)

Axis at x=0.5m

Torque (N-m)

Axis at =0.5m

F1

0.149

0.14602

0.351

0.34398

F2

0.211

0.41356

0.289

0.56644

F3

0.685

3.3565

0.185

-0.9065

F4

0.5

0.7005

0.5

0.7005

F5

0.091

-0.4186

0.409

-1.8814

F6

0.944

-4.248

0.444

1.998

Sum of Positive Torques

4.61658

3.60892

Sum of Negative Torques

-4.6666

-2.7879

%diff between positive and negative torques

5.002%

82.102%

Sum of all Torques

-0.05002

0.82102

Part II: Rotational Balance

The experiment uses a mass hanging on a string over a cylinder to exert a constant torque on the system resulting in a rotational balance of the system.

1. Measure the outer radius of the cylinder and the radius of the PVC fitting around which the string is wrapped.

2. Set up the apparatus as shown in Figure 3 and make sure the mass attached to the string is not touching either the board or the side of the cylinder. Set the angle of incline to 10o.

3. Place the string around the PVC fitting and make sure that it goes around the fitting at least for two turns.

4. Place enough mass at the end of the string so the cylinder does not roll down the incline. Record this mass as mmin

Figure 3

5. Start adding more mass to the mass found in the previous step until the cylinder starts rolling up the incline. Record this mass as mmax.

6. Find the average of mmin and mmax. Let σm= ( mmax - mmin )/2. Place all these values in the following table. Repeat the experiment for θ= 15 and 20 degrees.

Run

θºtheory

M (kg)

mmin

mmax

maverage

σm

θºexperiment

σθ

1

10

0.25697

0.1 kg

0.25 kg

0.175

0.075

0.02878

2

15

0.25697

0.15 kg

0.25 kg

0.2

0.05

0.031088

3

20

0.25697

0.4 kg

0.45 kg

0.425

0.025

0.044267

For your Lab Report:

Include a sample calculation of the torques, and % differences in Part I for x=0 condition. Find θexperiment±σθ for all 3 cases in Part II. To determine σθ use equation 6 and assume that the uncertainty for the numerator (A=mr) is 7%, and for the denominator (B=(M+m)R) is 4%. Compute θmax= sin-1[(A+σA)/(B-σB)] , θmin= sin-1[(A-σA)/(B+σB)] and σθ= (θmax - θmin )/2. Compare θexperiment to θtheory. Are they consistent? Within how many sigmas?

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