В этом latex коде поставь слева и справа от формул символ $. To address your task from the laboratory work, we'll focus on processing the measurement results, specifically on Task 1:

"На миллиметровой бумаге построить график зависимости периода колебаний ( T ) от ( g \cos \alpha ). Напиши какой формулой связаны ( T ) и ( g \cos \alpha ), какой вид будет иметь график."

Let's proceed step by step.

1. Deriving the Formula Connecting ( T ) and ( g \cos \alpha )
From the theoretical introduction provided in the lab work, the period ( T ) of small oscillations of a physical pendulum is given by equation (8):

[
T = 2\pi\sqrt{\dfrac{J}{m g_{\text{eff}} a}} = 2\pi\sqrt{\dfrac{l_{\text{пр}}}{g_{\text{eff}}}},
]

where:

( T ) is the period of oscillations,
( J ) is the moment of inertia of the pendulum about the axis of rotation,
( m ) is the mass of the pendulum,
( g_{\text{eff}} ) is the effective component of the acceleration due to gravity,
( a ) is the distance from the axis of rotation to the center of mass,
( l_{\text{пр}} = \dfrac{J}{m a} ) is the reduced (or equivalent) length of the physical pendulum.
In this experiment, the effective acceleration ( g_{\text{eff}} ) is given by:

[
g_{\text{eff}} = g \cos \alpha,
]

where:

( g ) is the acceleration due to gravity,
( \alpha ) is the angle between the plane of oscillation and the vertical plane.
Substituting ( g_{\text{eff}} ) into the expression for ( T ):

[
T = 2\pi\sqrt{\dfrac{l_{\text{пр}}}{g \cos \alpha}}.
]

This is the formula that connects the period ( T ) and ( g \cos \alpha ).

2. Understanding the Nature of the Graph
The formula:

[
T = 2\pi\sqrt{\dfrac{l_{\text{пр}}}{g \cos \alpha}}
]

shows that ( T ) is inversely proportional to the square root of ( g \cos \alpha ):

[
T \propto \dfrac{1}{\sqrt{g \cos \alpha}}.
]

When plotting ( T ) against ( g \cos \alpha ):

As ( g \cos \alpha ) increases, the denominator of the fraction inside the square root increases, making the entire fraction smaller.
Therefore, the square root of a smaller number decreases, and consequently, ( T ) decreases.
Graph Characteristics:

Shape: The graph of ( T ) versus ( g \cos \alpha ) will be a decreasing curve (non-linear), resembling a hyperbola in the first quadrant.
Behavior: It will start at a higher value of ( T ) when ( g \cos \alpha ) is small (i.e., when ( \alpha ) is large), and ( T ) will decrease as ( g \cos \alpha ) increases (i.e., as ( \alpha ) decreases).
Mathematical Nature: Since ( T ) is proportional to ( \dfrac{1}{\sqrt{g \cos \alpha}} ), the graph represents a portion of the function ( y = \dfrac{k}{\sqrt{x}} ), where ( k = 2\pi\sqrt{l_{\text{пр}}} ) is a constant.
Visualization Example:

To better understand the graph, let's consider sample values.

Sample Data Points:

Assuming ( l_{\text{пр}} ) and ( g ) are constants, and using different angles ( \alpha ):

( \alpha ) (degrees)	( \cos \alpha )	( g \cos \alpha ) (m/s²)	( T ) (s)
0	1.0000	( g \times 1 )	( T_0 )
15	0.9659	( g \times 0.9659 )	( T_1 )
30	0.8660	( g \times 0.8660 )	( T_2 )
45	0.7071	( g \times 0.7071 )	( T_3 )
60	0.5000	( g \times 0.5000 )	( T_4 )
75	0.2588	( g \times 0.2588 )	( T_5 )
90	0.0000	( 0 )	Undefined (since division by zero)
Observations:

At ( \alpha = 0^\circ ) (pendulum swinging in a vertical plane), ( \cos \alpha = 1 ), so ( g \cos \alpha = g ). The period ( T ) is at its minimum.
As ( \alpha ) increases towards ( 90^\circ ), ( \cos \alpha ) decreases to ( 0 ), and ( g \cos \alpha ) approaches ( 0 ), causing ( T ) to increase significantly.
At ( \alpha = 90^\circ ), ( \cos 90^\circ = 0 ), so ( g \cos \alpha = 0 ), and ( T ) becomes undefined (since we cannot divide by zero). Physically, this corresponds to the pendulum oscillating horizontally, which isn't possible under normal gravitational conditions.
Thus, the graph will show ( T ) starting at a finite value when ( g \cos \alpha ) is at its maximum (when ( \alpha = 0^\circ )), and ( T ) increasing (non-linearly) as ( g \cos \alpha ) decreases.

Conclusion
Formula Connecting ( T ) and ( g \cos \alpha ):

[
T = 2\pi\sqrt{\dfrac{l_{\text{пр}}}{g \cos \alpha}}.
]

Graph Description:

The graph of ( T ) versus ( g \cos \alpha ) is a decreasing curve.
It demonstrates an inverse square root relationship.
As ( g \cos \alpha ) increases, the period ( T ) decreases.
The graph resembles a portion of the function ( y = \dfrac{k}{\sqrt{x}} ).
Practical Implication:

When you plot your experimental data of ( T ) against ( g \cos \alpha ) on graph paper, you should expect to see this decreasing curve. This plot will help visualize how the period of oscillations depends on the effective component of gravity in the plane of oscillation.

Remember to label your axes clearly and include units. Also, note any deviations from the theoretical curve, which could be due to experimental errors or limitations.