Using simple mechanisms in physics allows us to study various natural processes and laws. One of these mechanisms is Atwood's machine. Consider in the article what it represents, what it is used for, and what formulas describe the principle of its operation.
What is Atwood Machine?
The named machine is a simple mechanism consisting of two weights that are connected by a thread (rope) thrown through a fixed block. In this definition, several nuances should be clarified. Firstly, the masses of cargo are generally different, which ensures that they have acceleration due to gravity. Secondly, the thread connecting the goods is considered weightless and inextensible. These assumptions greatly facilitate the subsequent calculations of the equations of motion. Finally, thirdly, the motionless block through which the thread is thrown is also considered weightless. In addition, friction is neglected during its rotation. The following diagram shows this machine.
Atwood's machine was invented by the English physicist George Atwood at the end of the 18th century. It serves to study the laws of translational motion, to accurately determine the acceleration of gravity and to experimentally verify Newton’s second law.
Dynamic equations
Every student knows that acceleration in bodies appears only if external forces act on them. This fact was established by Isaac Newton in the XVII century. The scientist stated it in the following mathematical form:
F = m * a.
Where m is the inertial mass of the body, a is the acceleration.
Studying the laws of translational motion on the Atwood machine involves knowledge of the corresponding equations of dynamics for it. Suppose that the masses of two goods are m 1 and m 2 , with m 1 > m 2 . In this case, the first load will move down under the action of gravity, and the second load will move up under the influence of the tension of the thread.
Consider what forces act on the first load. There are two of them: gravity force F 1 and tension force of the thread T. The forces are directed in different directions. Given the sign of acceleration a, with which the load moves, we obtain the following equation of motion for it:
F 1 - T = m 1 * a.
As for the second load, then forces of the same nature act on it as on the first. Since the second load moves with acceleration a directed upwards, the equation of dynamics for it takes the form:
T - F 2 = m 2 * a.
Thus, we wrote down two equations that contain two unknown quantities (a and T). This means that the system has a unique solution, which will be obtained later in the article.
Calculation of dynamic equations for uniformly accelerated motion
As we saw from the equations written above, the resulting force acting on each load remains unchanged during the entire movement. The mass of each cargo also does not change. This means that the acceleration a will be constant. Such a movement is called uniformly accelerated.
The study of uniformly accelerated motion on the Atwood machine is to determine this acceleration. We rewrite the system of dynamic equations:
F 1 - T = m 1 * a;
T - F 2 = m 2 * a.
To express the value of acceleration a, we add both equalities, we obtain:
F 1 - F 2 = a * (m 1 + m 2 ) =>
a = (F 1 - F 2 ) / (m 1 + m 2 ).
Substituting the explicit value of gravity for each load, we obtain the final formula for determining the acceleration:
a = g * (m 1 - m 2 ) / (m 1 + m 2 ).
The ratio of the mass difference to their sum is called the Atwood number. Denote it by n a , then we get:
a = n a * g.
Checking the solution of dynamics equations
Above, we determined the formula for accelerating Atwood's machine. It is valid only if Newton's law itself is valid. This fact can be verified in practice if laboratory work is carried out to measure certain quantities.
Laboratory work with the Atwood machine is quite simple. Its essence is as follows: as soon as the goods located at the same level from the surface are released, it is necessary to time the movement of goods with a stopwatch, and then measure the distance over which any of the goods has moved. Assume that the corresponding time and distance are t and h. Then we can write the kinematic equation of uniformly accelerated motion:
h = a * t 2/2.
From where the acceleration is determined uniquely:
a = 2 * h / t 2 .
Note that to increase the accuracy of determining the value of a, several experiments should be performed to measure h i and t i , where i is the measurement number. After calculating the values of a i , you should calculate the average value of a cp from the expression:
a cp = ∑ i = 1 m a i / m.
Where m is the number of measurements.
Equating this equality with that obtained earlier, we arrive at the following expression:
a cp = n a * g.
If this expression turns out to be true, then Newton’s second law will also be so.
Gravity calculation
We suggested above that the value of the acceleration of gravity g is known to us. However, using the Atwood machine, the determination of gravity is also possible. To do this, instead of accelerating a, from the equations of dynamics, the quantity g should be expressed, we have:
g = a / n a .
To find g, one should know what the acceleration of translational displacement is equal to. In the paragraph above, we have already shown how to find it experimentally from the kinematics equation. Substituting the formula for a into the equality for g, we have:
g = 2 * h / (t 2 * n a ).
By calculating the value of g, it is easy to determine the force of gravity. For example, for the first load, its value will be equal to:
F 1 = 2 * h * m 1 / (t 2 * n a ).
Determination of the thread tension
The force T of the thread tension is one of the unknown parameters of the system of dynamic equations. Let us write these equations again:
F 1 - T = m 1 * a;
T - F 2 = m 2 * a.
If in each equality to express a, and equate both expressions, then we get:
(F 1 - T) / m 1 = (T - F 2 ) / m 2 =>
T = (m 2 * F 1 + m 1 * F 2 ) / (m 1 + m 2 ).
Substituting the explicit values of the gravity of the cargo, we arrive at the final formula for the thread tension T:
T = 2 * m 1 * m 2 * g / (m 1 + m 2 ).
Atwood's machine has not only theoretical benefits. So, the elevator (elevator) uses a counterweight during its operation in order to lift the payload to a height. This design greatly facilitates the operation of the engine.