THE ATWOOD MACHINE (Newton's Second Law and the Conservation of Energy)

      

OBJECTIVE: To study the relation of masses and accelerations.

		FIGURE 1

 

METHOD:  Consider the Atwood machine shown in Fig. 1.  A pulley

is mounted on a support a certain distance above the floor.  A string

with loops on both ends is threaded through the pulley and different

weights are hung from both ends.  The lighter weight is placed near

the floor and the heavier weight near the pulley (the pulley can be

adjusted to the appropriate height).  The weights are then released

and the time for the weights to exchange places is measured.

Part 1.  Newton's Second Law

THEORY:  Consider the heavier weight, m₁.  There are two forces acting on it.  One is the force of gravity, W₁ = m₁g, pulling it downward.  The other force is the tension in the string, T₁, which is pulling the heavier weight upward.  Thus Newton's 2nd Law gives:

                                    SF_y = ‑m₁ g + T₁  = m₁ a₁ .                                                               (1)

Now consider the lighter weight, m₂.  Again there are two forces acting on it.  One is the force of gravity pulling it downward.  The other force is the tension in the string pulling it upward.  Thus Newton's 2nd Law gives:

                                    SF_y = ‑m₂ g + T₂  = m₂ a₂.                                                                (2)

Because the string is attached to both weights,  T₁ = T₂ = T, and a₂ = ‑a₁ = a.  From Eq. (2):

                                    T =  m₂ a + m₂ g  .                                                                             (3)

Substituting T from Eq. (3) into Eq. (1) gives:

                                    ‑m₁ g + (m₂ a + m₂ g)  =  ‑m₁ a

or                                 m₁ a + m₂ a  =  m₁ g ‑ m₂ g ,

or

                                    a  =   g (m₁ ‑ m₂ ) / (m₁ + m₂ )  .                                             (4)

By the appropriate choice of weights we can choose any acceleration we wish up to the value of g itself.  What choice of weights would give  a = 0?  What choice would give a = g?

PROCEDURE:

1)  Take m₁ = 50 gm and m₂ = 40 gm.  Allow m₁ to fall a distance h from near the pulley to the floor.  Measure the distance of the fall (make h = 100 cm by adjusting the pulley's position) and measure the time of the fall.  Repeat at least three times and take an average.

2) (a) Using the equations of motion for constant acceleration,

x = x_o + v_o t + ½ a t² ,   v = v_o + a t                                                     (5, 6)

determine the experimental acceleration and compare to the value predicted by Eq. (4).    (b) Are the theoretical and experimental accelerations the same? If not, is the uncertainty in the time measurement (usually 0.1 sec) sufficient to account for this difference?  That is, if you calculate the experimental acceleration using t^- = t_avg ‑ 0.1 sec and then calculate the experimental acceleration using t⁺ = t_avg + 0.1 sec., does the theoretical acceleration fall between these two values?  If the theoretical acceleration does not fall in this range, what else could account for this disagreement between theory and experiment?

3) Repeat Steps 1&2 for a machine with m₁ = 100 gm & m₂ = 90 gm.  Repeat once more for a machine with m₁ = 200 gm and m₂ = 190 gm.

REPORT:  Perform the calculations and answer the questions raised in the Procedure.

Part 2.  Conservation of Energy

THEORY:  We can consider the system conserving energy as consisting of both masses and the pulley.  The Law of Conservation of Energy states:

KE_1i + KE_2i +PE_1i + PE_2i + KE_pulley,i +W_T1 + W_T2 = KE_1f + KE_2f +PE_1f + PE_2f + KE_pulley,f + E_lost     (7)

Here, KE₁, KE₂, PE₁, PE₂ are the kinetic and potential energies of mass 1 and mass 2, KE_pulley is the kinetic energy of the pulley, W_T1 is the work done by tension on mass 1, W_T2 is the work done by tension on mass 2, and E_lost is the energy lost to friction.  Recall that kinetic energy is the energy of motion and that potential energy is the energy due to the position (height).  For the masses, KE = ½mv² and PE = mgh.  The kinetic energy of the pulley is the energy of its rotation.  We will see later in the course that this is given by KE = ½Iw² where I is called the ‘moment of inertia’ of the rotating object and w is its angular speed. The pulley starts from rest so KE_pulley,i = 0 and, because the pulley is small and light in this experiment, we will neglect its final kinetic energy.  Any transfer of initial energy to the pulley is small and will be included in the E_lost term along with energy lost to friction .  Also note that tension is an ‘internal’ force and does not contribute to the system's energy since any work done by the tension on the lighter mass is canceled by the work done by the tension on the heavier mass (W_T1 = -W_T2).  Thus, we can write the law of Conservation of Energy as

KE_1i + KE_2i +PE_1i + PE_2i = KE_1f + KE_2f +PE_1f + PE_2f + E_lost           (8)

PROCEDURE: 

1) (a) From the Law of Conservation of Energy, Eq. (8), determine what the theoretical final velocity should be for both masses if you neglect friction, that is, let E_lost = 0. 

(b) Now use the experimentally found time and the experimental value of the acceleration in the equation for constant acceleration, Eq. (6), to find the experimental final velocity of the masses.

(c) Are the theoretical and experimental final velocities the same? If not, can you explain why not? 

(d) Find the range for the experimental v. Be sure to use the acceleration calculated with t^- with the time t^- in Eq. (6) to find v^- , the upper limit for the velocity, and then use  the acceleration calculated with t⁺ with the time t⁺ in Eq. (6) to find v⁺, the lower limit for the velocity.

2)  If the theoretical and experimental accelerations do not agree within experimental uncertainty in time measurement, and if theoretical and experimental velocities do not agree within experimental uncertainty in the time measurement, friction may be blamed as the source of the error.  For those cases where there was not agreement between experiment and theory after allowing for uncertainty in time measurements, and thus where appreciable friction was present, calculate how much energy was lost to friction.  This can be done using the conservation of energy equation (Eq. (8)) with the friction term (E_lost) as the unknown with the experimental velocities, the masses, and the height as known.  NOTE:  Here you should obtain a range of possible energy losses rather than a single value because of the uncertainty in your velocity.  To get this range, calculate E_lost using v^- and again using v⁺.

REPORT:

1. Perform the calculations and answer the questions raised in the Procedure for EACH of the three machines

2.  Can you now state whether the energy lost to friction depends on the weight hung over the pulley?  Justify your answer.