Elasticity and heaviness
Goal of the work
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 on a thread moves in a circle. In this case, the thread describes a cone (Fig. 1). There are two forces acting on the ball: gravity and the elastic force of the thread. They create a centripetal acceleration directed radially 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 along 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 onto the selected axes (Fig. 2):
Oh: 
;
Oy: 
;
From the equation in 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 hence the acceleration: 
, where g=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
A tripod with a coupling and a foot, a measuring tape, a ball on a string, a sheet of paper with a drawn circle, a clock with a second hand
Progress
1. Hang the pendulum to 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 at the suspension point with your fingers and rotate the pendulum so that the ball describes a circle equal to the one drawn on the 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 modulus using the formulas:
8. Calculate errors.
Table Results of measurements and calculations
Computations
1. Period of circulation: 
; T=
2. Centripetal acceleration:
; a 1 =
; a 2 =
Average value of centripetal acceleration:
; and av =
3. Absolute error: 
∆a 1 =
∆a 2 =
4. Average absolute error: 
; Δa av =
5. Relative error: 
;
Conclusion
Record answers answer questions in complete sentences
1. Formulate the definition of centripetal acceleration. Write it down and the formula for calculating 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 calculation
gravity.
4. Write down the definition and formula for calculating the elastic force.
LAB WORK 5
Movement of a body at an angle to the horizontal
Target
Learn to determine the height and range of flight when moving a body with an initial speed directed at an angle to the horizon.
Equipment
Model “Motion of a body thrown at an angle to the horizontal” 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 horizontally (along the x-axis) and at the same time uniformly accelerated, with the acceleration of gravity, vertically (along the y-axis). This is how a skier moves when jumping from a springboard, a stream of water from a water cannon, artillery shells, throwing shells
Equations of motion s w:space="720"/>"> 
And 
Let's write in projections on the x and y axes:
To X axis: 
S= 
To determine the flight altitude, it is necessary to remember that at the top point of ascent the body speed is 0. Then the ascent time will be determined: 
When falling, the same amount of time passes. Therefore, the movement 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 horizontal.xls”.
3. In cell B2 enter the value of the initial speed, 15 m/s, and in cell B4 - 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 values to 25 m/s and the angle to –35 degrees; 18 m/s, 55 degrees. Review the graphs.
6. Perform formula calculations for speed and angle values(according to options):
8. Check your results, look at the graphs. Draw the graphs to scale on a separate A4 sheet
Table Values of sines and cosines of some angles
| 30 0 | 45 0 | 60 0 | |
| Sine (Sin) | 0,5 | 0,71 | 0,87 |
| Cosine (Cos) | 0,87 | 0,71 | 0,5 |
Conclusion
Write down the answers to the questions in complete sentences
1. On what values 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 horizontal.
3. At what angle to the horizon is the greatest range of flight of a body observed at an angle to the horizon?
LAB 6
The study of the movement of a body in a circle under the influence of elasticity and gravity.
Purpose of the work: determination of the centripetal acceleration of a ball during its uniform motion in a circle.
Equipment: a tripod with a coupling and a foot, a measuring tape, a compass, a laboratory dynamometer, a scale with weights, a ball on a string, a piece of cork with a hole, a sheet of paper, a ruler.
1. Let’s bring the load into rotation along a drawn circle of radius R= 20 cm. We measure the radius with an accuracy of 1 cm. Let’s measure the time t during which the body will make N=30 revolutions.
2. Determine the vertical height h of the conical pendulum from the center of the ball to the suspension point. h=60.0 +- 1 cm.
3. 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 F1 F1 = 0.12 N, the mass of the ball m = 30 g + - 1 g.
4. We enter the measurement results into a table.
5.Calculate an using the formulas given in the table.
6.The result of the calculation is entered into the table.
Conclusion: comparing the obtained three values of the centripetal acceleration module, we are convinced that they are approximately the same. This confirms the correctness of our measurements.
3. Calculate and enter into the table the average value of the time period<t> for which the ball makes N= 10 revolutions.
4. Calculate and enter into the table the average value of the rotation period<T> ball.
5. Using formula (4), determine and enter into the table the average value of the acceleration modulus.
6. Using formulas (1) and (2), determine and enter into the table the average value of the angular and linear velocity modules.
| Experience | N | t | T | a | ω | v |
| 1 | 10 | 12.13 | — | — | — | — |
| 2 | 10 | 12.2 | — | — | — | — |
| 3 | 10 | 11.8 | — | — | — | — |
| 4 | 10 | 11.41 | — | — | — | — |
| 5 | 10 | 11.72 | — | — | — | — |
| Wed. | 10 | 11.85 | 1.18 | 4.25 | 0.63 | 0.09 |
7. Calculate the maximum value of the absolute random error in measuring the time interval t.
8. Determine the absolute systematic error of the time period t .
9. Calculate the absolute error of direct time measurement t .
10. Calculate the relative error of direct measurement of the time interval.
11. Write down the result of direct measurement of a period of time in interval form.
Answer security questions
1. How will the linear speed of the ball change when it rotates uniformly relative to the center of the circle?
Linear speed is characterized by direction and magnitude (modulus). The modulus is a constant quantity, but the direction during such movement can change.
2. How to prove the ratio v = ωR?
Since v = 1/T, the relationship between the cyclic frequency and the period is 2π = VT, whence V = 2πR. The connection between linear velocity and angular velocity is 2πR = VT, hence V = 2πr/T. (R is the radius of the described, r is the radius of the inscribed)
3. How does the rotation period depend? T ball from the module of its linear velocity?
The higher the speed indicator, the lower the period indicator.
Conclusions: learned to determine the period of rotation, modules, centripetal acceleration, angular and linear speeds during uniform rotation of a body and calculate the absolute and relative errors of direct measurements of the time interval of body movement.
Super task
Determine the acceleration of a material point during its uniform rotation, if for Δ t= 1 s she covered 1/6 of the circumference, having a modulus of linear velocity v= 10 m/s.
Circumference:
S = 10 ⋅ 1 = 10 m
l = 10⋅ 6 = 60 m
Circle radius:
r = l/2π
r = 6/2 ⋅ 3 = 10 m
Acceleration:
a = v 2/r
a = 100 2 /10 = 10 m/s2.
“Study of the motion of a body in a circle under the action of two forces”
Goal of the work: determination of the centripetal acceleration of a ball during its uniform motion in a circle.
Equipment: 1. tripod with coupling and foot;
2. measuring tape;
3. compass;
4. laboratory dynamometer;
5. scales with weights;
6. ball on a string;
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 pass the thread through the hole and clamp the plug in the tripod foot (Fig. 1)
3. Draw a circle on a piece 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 the paper.
6. We count the time during which the pendulum makes, for example, N=50 revolutions. Calculating 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. Using a horizontal dynamometer, we pull the ball 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. We enter the measurement results into a table.
| Experience no. | 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 in the form: a n 1 = a n 1av ± ∆ a n 1av a n 1 =
Draw a conclusion:
Control questions:
1. What type of motion is the movement of a ball on a string in laboratory work? Why?
2. Make a drawing in your notebook and indicate the names of the forces correctly. Name the points of application of these forces.
3. What laws of mechanics are satisfied when the body moves in this work? Draw graphically the forces and write the laws correctly
4. Why is the elastic force F, measured experimentally, equal to the resultant forces applied to the body? Name the law.
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