Dynamics of rotational motion. Determination of loads inertia moment using oberbech’s pendulum

The purpose of the work. to study the basic rotational motion dynamics law; to determine the rotating solid body inertia moment.

Devices and accessories 1. Pendulum. 2. Set of loads. 3. Electronic stop-watch. 4. Ruler.

Task 1. Determination of inertia moment of Oberbech’s pendulum crosspiece.

1.1. Remove loads from the rods.

1 .2. Measure the diameter of pulley with trammel. Write down the result as:

__________________ m,

where R_in is instrumental error of radius determination.

1.3. Write down the value of falling load mass and height, the load falls from:

__________________ kg,

__________________ m,

where m is an instrumental error of mass measurement; h is an instrumental error of height measurement.

1.4. Let the load go down and measure the time of its falling down to a lower score with a stop watch. Repeat measurement 5 times and enter the results into table 1.

Task 2. Determination of inertia moment of Oberbech’s pendulum with loads

2.1. Fix loads at the first rods scores near the pulley and carry out measurements according to item 1.4.

2.2. Determine the distance between the centre of rotating loads and the centre of pendulum rotation. Enter the result into table 1.

2.3. Fix loads on a medial, and then on farthest scores of the rods and repeat measurements according to items 1.4 and 2.2. Enter the obtained result into table 1.

2.4. Determine random error of falling load time measurement for position without loads:

— calculate average values

where n = 5 is amount of measurement;

— calculate the deviation of the measured values of time from it mean value

— determine the root-mean-square error:

— calculate the absolute error of falling load time measurement:

where the Student’s coefficient is t_St = 2,78 (n = 5 and  = 0,95);

— determine the absolute error according to the formula:

t = ,

— determine the fractional error according to the formula:

Enter the results into table 2.

Table 1

Position	Exp. No	Time of the load falling down, t, s	Average value of time, , s	Pulley radius, R, m	Distance from the center of rotation to the load center m	Height, the load falls from h, m	Mass of the falling load, m, kg
Without loads	1				0
	2
	3
	4
	5
І	1
	2
	3
	4
	5
ІІ	1
	2
	3
	4
	5
ІІІ	1
	2
	3
	4
	5

Table 2

Position	Exp. No	Average value tave, s	Deviation from mean value s	Root-mean-square deviation s	Absolute error t, s	Fractional error t, %
Without loads	1
	2
	3
	4
	5

Task 3. Determination of inertia moments of the loads in three positions

3.1. Calculate the moments of inertia of pendulum without loads and with loads in positions I, II and III:

= kgm²,

= kgm²,

= kgm²,

= kgm².

Enter the results into table 3.

3.2. Calculate the fractional error of the inertia moment of pendulum measurement using formula:

_p = =

=_______________________________________________________________________________

where m_in, h_in, R_in are instrumental errors of load mass, height and pulley radius measurements; t is absolute error of load falling down time measurement.

3.3. Calculate the absolute error of determination using formula:

=______________________________________________ kgm².

Task 4. Graph plotting.

Plot the graph as the dependence of loads inertia moment upon the square of loads distance to the rotation centre.

4.1. Calculate inertia moments of the loads in three positions.

kgm²,

kgm²,

kgm².

Enter the results into table 3.

Table 3

Position	Distance from the center of rotation to the load center m	Moment of inertia, I, kg · m2			, m2
ithout loads	0
I
II
III

4.2. Plot the graph as the dependence of loads inertia moment upon the square of loads distance to the rotation centre, using data from Table 3.

C onclusions

The data of laboratory work fulfillment

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questions to be admitted for doing laboratory work and its defending

1. Give the definition of rotational motion.

2. Name kinematic characteristics of rotational motion

3. Name dynamic characteristics of rotational motion.

4. What is moment of force (torque)? What does moment of force depend on? Formulate the rule of determination of torque direction.

5. What is angular momentum? What does angular moment depend on?

6. What is inertia moment? What are the equations applied for determination of inertia moment of material point and rigid body?

7. Formulate Steiner’s theorem and explain its application.

8. Formulate basic dynamics laws for translational motion of material point and rigid body.

9. Formulate basic dynamics laws for rotational motion of material point and rigid body.

10. Formulate the angular momentum conservation law.

11. Deduce the expression for calculation of moment of inertia.

12. Formulate the main stages of laboratory work fulfillment.

LABORATORY WORK

DETERMINATION OF LIQUID VISCOSITY COEFFICIENT

The purpose of the work: to study the transport phenomena; to determine the internal friction coefficient by the Stock’s method

Devices and accessories Glass cylinder with glycerin. Metal and aluminum balls of different diameters. Stop-watch. Microscope МПВ-2.

Task 1. Determination of time of balls falling down

1. Take 5 balls.

2. Measure diameter of each ball а_i by means a number of peaches of microscope МПВ-2. Repeat measurements 3 times. Calculate average values a_i ave of the ball diameter. Enter the obtained results into table 1.

Table 1

Exp. No	Ball diameter, аi, pitchers	Average value of ball diameter, ai ave, pitchers	Scale factor of microscope, N, mm	Average value of ball radius, ri ave, m	Height, the ball falls from l, m	Time, the ball falling down ti, s	Viscosity,
The first ball
1
2
3
The second ball
1
2
3
The third ball
1
2
3
The forth ball
1
2
3
The fifth ball
1
2
3

1.3. Calculate average value of ball radius r_i ave for each ball according to the formula:

where n = 3 is amount of measurements. Enter the results into table 1.

1.4. Measure the distance between the upper and the lower scores l of the cylinder by a ruler. Enter the results into table 1.

1.5. Take the first ball with known radius and fall it down in liquid. Measure the time the ball takes to fall the distance l. Repeat measurement with all other balls. Enter the results into table 1.

1.6. Measure the liquid temperature by a thermometer. Enter the results into table 2.

1.7. Enter values of material density  the balls made of and liquid density _l into table 2.

1.8. Calculate the internal friction coefficient for each ball according to the formula:

.

Enter the results into table 1.

Table 2

Liquid temperature, Tl, К	Material density the balls made of, , kg\m3	Liquid density, l, kg/m3

Task 2. Calculation of viscosity measurement error

Calculate average value of the viscosity according to the formula:

where n = 5 is amount of measurements. Enter the results into table 3.

2.2. Calculate deviation from average value Δη_i according to the formula:

Table 3

Exp. No	Viscosity	Average value of vicosity ,	Deviation of mean value ,	Root-mean-square deviation	Absolute error, ,	Fractional error,
1
2
3
4
5

2.3. Calculate root-mean-square error of the arithmetic mean according to the formula:

2.4. Determine the viscosity measurement absolute error according to the formula:  = t_St .

where the Student’s coefficient is t_St = 2,78 (n = 5 and  = 0,95)

2.5. Calculate the viscosity measurement fractional error according to the formula:

2.6. Write down the result in a form: ____________________________,

Conclusions

The data of laboratory work fulfillment

Pass mark Signature

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