We know from the textbook (pp. 15-16) that when moving uniformly in a circle, the speed of a particle does not change in magnitude. In fact, from a physical point of view, this movement is accelerated, since the direction of the velocity is continuously changing in time. In this case, the speed at each point is practically directed along the tangent (Fig. 9 in the textbook on page 16). In this case, acceleration characterizes the rate of change in the direction of velocity. It is always directed towards the center of the circle along which the particle moves. For this reason, it is commonly called centripetal acceleration.
This acceleration can be calculated using the formula:
The speed of movement of a body in a circle is characterized by the number of complete revolutions per unit of time. This number is called the rotational speed. If the body makes v revolutions per second, then the time it takes to complete one revolution is
seconds. This time is called the rotation period.
To calculate the speed of a body in a circle, you need the path traveled by the body in one revolution (it is equal to the length
circles) divided by the period:
in this work we
we will observe the movement of a ball suspended on a thread and moving in a circle.
An example of a job.
4.2.1. Prepare the scales and, with the permission of the laboratory assistant, weigh the body. Determine the instrumental error of the scales.
4.2.2. Record the measurement result in standard form: m=(m±Δm) [dimension].
5. CONCLUSION
Indicate whether the goal of the work has been achieved.
Record body weight measurements in two ways.
5.3. Compare results. Draw a conclusion
6. CONTROL QUESTIONS
6.1. What is inertial mass, gravitational mass, how are they defined? Formulate the principle of equivalence of inertial and gravitational mass.
6.2. What are direct measurements and indirect measurements? Give examples of direct and indirect measurements.
6.3. What is the absolute error of the measured quantity?
6.4. What is the relative error of the measured quantity?
6.5. What is the confidence interval of the measured quantity?
6.6. List the types of errors and give a brief description of them.
6.7. What is the accuracy class of the instrument? What is the price division of the device?
How is the instrumental error of the measurement result determined?
6.8. How the relative error and absolute error of indirect measurement are calculated.
6.9. How is the standard recording of the final measurement result made? What requirements must be met?
6.10. Measure the linear size of the body with a caliper. Record the measurement result in standard form.
6.11. Measure the linear size of the body with a micrometer. Record the result.
Laboratory work №2.
The study of the movement of the body in a circle
1. PURPOSE OF THE WORK. Determination of the centripetal acceleration of a ball during its uniform motion in a circle.
2. INSTRUMENTS AND ACCESSORIES. A tripod with a clutch and a foot, a ruler, a tape measure, a ball on a thread, a sheet of paper, a stopwatch.
BRIEF THEORY
The experiment is carried out with a conical pendulum (Fig. 1). Let a ball suspended on a thread describe a circle with a radius R. There are two forces acting on the ball: gravity and tension in the string. Their resultant creates a centripetal acceleration directed towards the center of the circle. The acceleration modulus can be determined using kinematics:
(1)
To determine the acceleration, it is necessary to measure the radius of the circle R and the period T circulation of the ball around the circle.
Centripetal acceleration can also be determined using Newton's 2nd law:
We choose the direction of the coordinate axes as shown in Fig.1. We project equation (2) onto the selected axes:
From equations (3) and (4) and from the similarity of triangles we get:
Fig.1. 
. (5)
Thus, using equations (1), (3) and (5), centripetal acceleration can be determined in three ways:
. (6)
Component module F x can be directly measured with a dynamometer. To do this, we pull the ball with a horizontally located dynamometer to a distance equal to the radius R circle (Fig. 1), and determine the dynamometer reading. In this case, the elastic force of the spring balances the horizontal component F x and equal in size.
In this work, the task is to verify experimentally that the numerical values of the centripetal acceleration obtained in three ways will be the same (the same within the absolute errors).
WORK TASK
1. Determine the mass m balls on the scales. Weighing result and instrumental error ∆ m write in table 1.
2. We draw a circle with a radius of about 20 cm on a piece of paper. We measure this radius, determine the instrumental error, and write the results in table 1.
3. Position the tripod with the pendulum so that the continuation of the thread passes through the center of the circle.
4. Take the thread with your fingers at the point of suspension and rotate the pendulum so that the ball describes the same circle as the circle drawn on paper.
5. Counting the time t, for which the ball makes a given number of revolutions (for example, N= 30) and estimate the error ∆ t measurements. The results are recorded in table 1.
6. Determine the height h conical pendulum and instrumental error ∆ h. Distance h measured vertically from the center of the ball to the point of suspension. The results are recorded in table 1.
7. We pull the ball with a horizontally located dynamometer to a distance equal to the radius R of the circle, and determine the dynamometer reading F= F x and instrumental error ∆ F. The results are recorded in table 1.
Table 1.
| m | ∆m | R | ∆R | t | ∆t | N | h | ∆h | F | ∆F | g | ∆g | π | ∆ π |
| G | G | mm | mm | With | With | mm | mm | H | H | m/s 2 | m/s 2 | |||
8. Calculate the period T circulation of the ball around the circle and the error ∆ T:
.
9. Using formulas (6), we calculate the values of centripetal acceleration in three ways and the absolute errors of indirect measurements of centripetal acceleration.
CONCLUSION
In the output, write in standard form the values of centripetal acceleration obtained in three ways. Compare the obtained values (see section "Introduction. Measurement errors"). Make a conclusion.
TEST QUESTIONS
6.1. What is a period T
6.2. How can you experimentally determine the period T the circle of the ball?
6.3. What is centripetal acceleration, how can it be expressed in terms of the period of revolution and in terms of the radius of the circle?
6.4. What is a conical pendulum. What forces act on the ball of a conical pendulum?
6.5. Write down Newton's 2nd law for a conical pendulum.
6.6. What are the three methods for determining centripetal acceleration offered in this lab?
6.7. What measuring devices are used to determine the values of the physical quantities given in Table 1?
6.8. Which of the three methods for determining centripetal acceleration gives the most accurate value of the measured quantity?
Lab #3
Similar information.
For grade 9 (I.K. Kikoin, A.K. Kikoin, 1999),
a task №5
to chapter " LABORATORY WORKS».
The purpose of the work: to make sure that when a body moves in a circle under the action of several forces, their resultant is equal to the product of the body's mass and acceleration: F = ma . For this, a conical pendulum is used (Fig. 178, a).
On the body attached to the thread (in the work it is a load from
set in mechanics) the force of gravity F 1 and the force of elasticity F 2 act. Their resultant is
Force F and imparts centripetal acceleration to the load
(r is the radius of the circle along which the load moves, T is the period of its revolution).
To find the period, it is convenient to measure the time t of a certain number N of revolutions. Then T =
The resultant modulus F of the forces F 1 and F 2 can be measured by compensating it with the elastic force F of the dynamometer spring, as shown in Figure 178, b.
According to Newton's second law,
When substituting into
this is the equality of the values F ynp , m and a obtained in the experiment, it may turn out that the left side of this equality differs from unity. This allows us to estimate the error of the experiment.
Measuring instruments: 1) ruler with millimeter divisions; 2) clock with a second hand; 3) dynamometer.
Materials: 1) tripod with sleeve and ring; 2) strong thread; 3) a sheet of paper with a circle drawn with a radius of 15 cm; 4) a load from the mechanics kit.
Work order
1. Tie a thread about 45 cm long to the weight and hang it from the tripod ring.
2. For one of the students, grab the thread at the suspension point with two fingers and rotate the pendulum.
3. For the second student, measure the radius r of the circle along which the load moves with a tape. (A circle can be drawn in advance on paper and a pendulum can be set in motion along this circle.)
4. Determine the period T of the pendulum using a clock with a second hand.
To do this, the student rotating the pendulum, in time with its revolutions, says aloud: zero, zero, etc. The second student with a clock in his hands, catching the convenient moment to start the countdown in the second hand, says: “zero”, after which the first student aloud counts the number of revolutions. After counting 30-40 revolutions, fixes the time interval t. The experiment is repeated five times.
5. Calculate the average value of acceleration using formula (1), considering that with a relative error of not more than 0.015, π 2 = 10 can be considered.
6. Measure the modulus of the resultant F, balancing it with the elastic force of the dynamometer spring (see Fig. 178, b).
7. Enter the measurement results in the table:
8. Compare the ratio
with unity and draw a conclusion about the error of the experimental verification that the centripetal acceleration informs the body of the vector sum of the forces acting on it.
A load from the mechanics set, suspended on a thread fixed at the top point, moves in a horizontal plane along a circle of radius r under the action of two forces:
gravity
and elastic force N .
The resultant of these two forces F is directed horizontally to the center of the circle and imparts centripetal acceleration to the load.
T is the period of circulation of the cargo around the circumference. It can be calculated by counting the time for which the load makes a certain number of complete revolutions.
Centripetal acceleration is calculated by the formula
Now, if we take a dynamometer and attach it to the load, as shown in the figure, we can determine the force F (the resultant of the forces mg and N.
If the load is deflected from the vertical by a distance r, as in the case of movement in a circle, then the force F is equal to the force that caused the movement of the load in a circle. We get the opportunity to compare the value of the force F obtained by direct measurement and the force ma calculated from the results of indirect measurements and
compare ratio
with unit. In order for the radius of the circle along which the load moves to change more slowly due to the influence of air resistance and this change slightly affects the measurements, it should be chosen small (of the order of 0.05 ~ 0.1 m).
Completing of the work
Computing
Estimation of errors. Measurement accuracy: ruler -
stopwatch
dynamometer
We calculate the error in determining the period (assuming that the number n is determined exactly):
The error in determining the acceleration is calculated as:
Error in determining ma
(7%), that is
On the other hand, we measured the force F with the following error:
This measurement error is, of course, very large. Measurements with such errors are only suitable for rough estimates. It can be seen from this that the deviation
from unity can be significant when using the measurement methods used by us * .
1 * So you should not be embarrassed if in this lab the ratio
will be different from unity. Just carefully evaluate all the measurement errors and draw the appropriate conclusion.
"The study of the motion of a body in a circle under the action of two forces"
Objective: determination of the centripetal acceleration of a ball during its uniform motion in a circle.
Equipment: 1. tripod with clutch and foot;
2. measuring tape;
3. compass;
4. laboratory dynamometer;
5. scales with weights;
6. ball on a thread;
7. a piece of cork with a hole;
8. sheet of paper;
9. ruler.
Work order:
1. Determine the mass of the ball on the scales with an accuracy of 1 g.
2. We thread the thread through the hole and clamp the cork in the foot of the tripod (Fig. 1)
3. We draw a circle on a sheet of paper, the radius of which is about 20 cm. We measure the radius with an accuracy of 1 cm.
4. We position the tripod with the pendulum so that the extension of the cord passes through the center of the circle.
5. Taking the thread with your fingers at the suspension point, rotate the pendulum so that the ball describes a circle equal to that drawn on paper.
6. We count the time during which the pendulum makes, for example, N=50 revolutions. We calculate the circulation period T=
7. Determine the height of the conical pendulum. To do this, measure the vertical distance from the center of the ball to the suspension point.
8. Find the modulus of normal acceleration using the formulas:
a n 1 = a n 2 =
a n 1 = a n 2 =
9. We pull the ball with a horizontally located dynamometer to a distance equal to the radius of the circle, and measure the modulus of the component F
Then we calculate the acceleration using the formula a n 3 = a n 3 =
10. The results of measurements are entered in the table.
| experience number | R m | N | ∆t c | T c | h m | m kg | F N | a n1 m/s 2 | a n 2 m/s 2 | a n 3 m/s 2 |
Calculate the relative calculation error a n 1 and write the answer as: a n 1 = a n 1av ± ∆ a n 1av a n 1 =
Conclude:
Test questions:
1. What type of movement is the movement of a ball on a thread in laboratory work? Why?
2. Make a drawing in your notebook and indicate the correct names of the forces. Name the points of application of these forces.
3. What laws of mechanics are fulfilled when the body moves in this work? Draw graphically the forces and write down the laws correctly
4. Why is the elastic force F, measured in the experiment, equal to the resulting forces applied to the body? Name the law.
 |
Elasticity and gravity
Objective
Determination of the centripetal acceleration of a ball during its uniform motion in a circle
Theoretical part of the work
Experiments are carried out with a conical pendulum: a small ball suspended from a thread moves in a circle. In this case, the thread describes a cone (Fig. 1). Two forces act on the ball: the force of gravity and the force of elasticity of the thread. They create a centripetal acceleration directed along the radius towards the center of the circle. The acceleration modulus can be determined kinematically. It is equal to:
To determine the acceleration (a), you need to measure the radius of the circle (R) and the period of revolution of the ball around the circle (T).
Centripetal acceleration can be determined in the same way using the laws of dynamics.
According to Newton's second law, 
Let's write this equation in projections on the selected axes (Fig. 2):
Oh: 
;
Oy: 
;
From the equation in the projection onto the Ox axis, we express the resultant: 
From the equation in projection onto the Oy axis, we express the elastic force: 
Then the resultant can be expressed: 
and here is the acceleration: 
, where g \u003d 9.8 m / s 2
Therefore, to determine the acceleration, it is necessary to measure the radius of the circle and the length of the thread.
Equipment
Tripod with clutch and claw, measuring tape, ball on a thread, a sheet of paper with a drawn circle, a clock with a second hand
Progress
1. Hang the pendulum from the tripod leg.
2. Measure the radius of the circle with an accuracy of 1mm. (R)
3. Position the tripod with the pendulum so that the extension of the cord passes through the center of the circle.
4. Take the thread with your fingers at the suspension point, rotate the pendulum so that the ball describes a circle equal to that drawn on paper.
6. Determine the height of the conical pendulum (h). To do this, measure the vertical distance from the suspension point to the center of the ball.
7. Find the acceleration module using the formulas:
8. Calculate the errors.
Table Results of measurements and calculations
Computing
1. Period of circulation: 
; T=
2. Centripetal acceleration:
; a 1 =
; a 2 =
Average value of centripetal acceleration:
; a cp =
3. Absolute error: 
∆a 1 =
∆a 2 =
4. Average value of absolute error: 
; Δа ср =
5. Relative error: 
;
Conclusion
Record responses questions in full sentences
1. Formulate the definition of centripetal acceleration. Write it down and the formula for calculating the acceleration when moving in a circle.
2. Formulate Newton's second law. Write down its formula and wording.
3. Write down the definition and formula for calculating
gravity.
4. Write down the definition and formula for calculating the elastic force.
LAB 5
Body movement at an angle to the horizon
Target
Learn to determine the height and range of flight when the body moves with an initial speed directed at an angle to the horizon.
Equipment
Model "Movement of a body thrown at an angle to the horizon" in spreadsheets
Theoretical part
The movement of bodies at an angle to the horizon is a complex movement.
Movement at an angle to the horizon can be divided into two components: uniform movement along the horizontal (along the x axis) and simultaneously uniformly accelerated, with free fall acceleration, along the vertical (along the y axis). This is how a skier moves when jumping from a springboard, a jet of water from a hose, artillery shells, projectiles
Equations of motion s w:space="720"/>"> 
and 
we write in projections on the x and y axes:
For X-axis: 
S= 
To determine the flight altitude, it must be remembered that at the top point of the ascent, the speed of the body is 0. Then the ascent time will be determined: 
When falling, the same time passes. Therefore, the travel time is defined as
Then the lift height is determined by the formula:
And the flight range:
The greatest flight range is observed when moving at an angle of 45 0 to the horizon.
Progress
1. Write down the theoretical part of the work in your workbook and draw a graph.
2. Open the file "Movement at an angle to the horizon.xls".
3. In cell B2, enter the value of the initial speed, 15 m/s, and in cell B4, enter the angle of 15 degrees(only numbers are entered in the cells, without units of measurement).
4. Consider the result on the graph. Change the speed value to 25 m/s. Compare Graphs. What changed?
5. Change the speed to 25 m/s and the angle to -35 degrees; 18 m/s, 55 degrees. Consider charts.
6. Perform formula calculations for speeds and angles(by options):
8. Check your results, look at the graphs. Draw graphs to scale on a separate A4 sheet
Table Values of sines and cosines of some angles
| 30 0 | 45 0 | 60 0 | |
| Sinus | 0,5 | 0,71 | 0,87 |
| Cosine (Cos) | 0,87 | 0,71 | 0,5 |
Conclusion
Write down the answers to the questions complete sentences
1. On what quantities does the flight range of a body thrown at an angle to the horizon depend?
2. Give examples of the movement of bodies at an angle to the horizon.
3. At what angle to the horizon is the greatest range of flight of the body at an angle to the horizon?
LAB 6
