Self-examination questions

1. How is the histogram of the measured values constructed?

2. Explain the essence of a distribution function.

3. Describe the procedure of hypothesis about random distribution normality verification.

4. Deduce a differential equation of a mathematical pendulum motion.

5. Write down the equation of free undamped harmonic oscillations and analyse it.

6. Deduce the expression for determination of gravity acceleration using mathematical pendulum.

7. Derive the formulas for calculation of fractional and absolute errors of gravity acceleration determination.

Laboratory work 1.3

Rotational motion laws study and a moment of inertia determination

The purpose of the work: experimental determination of a rotary solid inertia moment.

Theoretical information: before doing the work, you are to study theoretical material on dynamics of rotational motion. Measurement procedure and experimental equipment

I n this work determination of rotating solids inertia moment is carried out by means of a cruciform pendulum (Oberbech’s pendulum), shown in Fig. 5.5. It is an arbour 5 with pulley 1, rotating on bearings. In one plane perpendicularly to the arbour, four rods 3 are located, on which bodies 2 of cylindrical shape are fixed. A rope with adhered load 6 is reeled on the pulley.

D

Fig. 2.1

ue to gravity acting on the load, the rope is unreeled, which causes the uniformly accelerated gyration of the pendulum.

The inertia moment of any rotary solid can be determined by the basic law of rotational motion:

Fig. 5.5 .

From the formula, it is evident that, for experimental determination of rotary system inertia moment, it is necessary to find the torque causing gyration of this system and the angular acceleration which the system gains under this torque action.

The rotary acceleration can be determined by using its connection with a linear acceleration:

,

where R is the radius of the pulley.

As the load falls down from height h during time t, its acceleration is equal to

.

It is evident that all points of the pulley rim have the same acceleration. Hence, the acceleration is the pulley rim tangential acceleration. As a result, the angular acceleration is equal to

.

Using the preceding equations we obtain

.

The torque of the system is formed by tension Т and is equal to

.

For the uniformly accelerated motion of a load, the dynamical equation looks as follows:

, so .

Taking into account the previous expressions, we get the equation

.

Then the moment of rope tension force about the axis of rotation is equal to:

.

Using the basic law of dynamics, we get the expression for calculation of inertia moment

. (5.8)

To find the error of indirect measurements of the moment of inertia of Oberbech’s pendulum, let’s take a look at formula (5.8). In the formula there are four values which are measured directly, namely: m, R, t, h. They are partly exponential functions, partly – algebraic sum. In that case, the best is to begin with finding the moment of inertia relative error. First we take the logarithm of the formula

.

Then we find partial derivative of the moment of inertia with respect to mass, radius, height and time:

; ;

;

Having substituted expressions for derivatives in the general formula

, we receive the expression for a relative error of the moment of inertia

(5.9)

where m, R, t, h – absolute errors of direct measurements. After that, it is possible to determine the absolute error of the moment of inertia

І = ε_I·I. (5.10)