Brown University Physics Department
Physics 0050/0070
ACCELERATION ACCELERATION OF A FALLING YO-YO (L-15) THEORY:
A yo-yo is made of two identical disks attached at their centers to a cylindrical axle, which has a smaller radius. A string is attached to and wrapped around this axle. If the yo-yo is released, it displays both downward linear motion and circular motion about the axle. To understand the motion of the yo-yo, let us denote the radius of the disks as r , the radius of the axle as b (note: the radius of the axle does do es change slightly as the string unwinds; however, this is a fairly small change and for the sake of simplicity we will consider it to be a constant), the mass of the yo-yo as M , and the moment of inertia as I as I cm cm. In some textbooks I textbooks I cm cm is assumed to be roughly equal to that for a uniform disk, but we will just denote it as I as I cm cm for now; the equation will be given later.
T d
Tension d
b
D
T
r
r
Mg
b
Weight
Side View
Front View
When the yo-yo falls, there are two forces acting on it: tension along the string in the upward direction, and gravity in the downward direction. All forces and linear motion then are along the vertical, and the net force gives us gravity minus tension if down is the positive direction. This gives the equation Mg-T=Ma y
(1)
for linear motion. For the angular motion, the torque is solely dependent on the tension along the string—gravity doesn’t contribute. The equation for angular acceleration is therefore Tb=I cm cmα
(2)
Brown University Physics Department
Physics 0050/0070
where α is the angular acceleration and therefore equal to the linear acceleration, a y a y divided by the radius, or . We can substitute this value for the angular acceleration b and get the equation ay Tb = I cm (3) b which puts both equation 1 and 3 in terms of T and a. Eliminating T in equation 1 through substitution gives the equation I Mg − cm2 a y = Ma y b
(4)
which can in turn be solved for the acceleration, giving a y =
Mg M + I cm
⎛ ⎞ 1 ⎜ g = ⎜ 1 + I Mb 2 ⎟⎟ < g . b2 cm ⎝ ⎠
(5)
This can be substituted back into one of the earlier equations to solve for the tension:
⎛ ⎞ 1 ⎜ ⎟⎟ < Mg . T = Mg Mg = 2 2 ⎜ 1 + I cm Mb ⎝ Mb I cm + 1 ⎠ I cm b 2
(6)
The moment of inertia can be given as the sum of the moments of inertia of each of the end disks plus the moment of inertia of the axle: 1 ⎛ 1 ⎞ I cm = M 1b 2 + 2⎜ M 2 r 2 ⎟ (7) 2 2 ⎝ ⎠ where M 1 is the mass of the axle and M 2 the mass of each disk. Substitute this into the values for a y and T to get the answer. The kinetic energy of the yo-yo can be determined using the conservation of energy. The top end of the string doesn’t move and so the string does no work on the yoyo. Gravity does positive work in the y direction. No energy is added to the system, so the decrease in potential energy corresponding to the decrease in height is equal to the increase in kinetic energy, Mgh. However, kinetic energy is not merely linear here: it is both translational (in this case, the linear kinetic energy) and rotational: 1 1 2 K net = Mv y + I cmω 2 (8) 2 2 where v is the velocity and ω the angular velocity. The angular velocity is equal to the linear velocity divided by the radius (the radius of the axle, in this case, or b), which gives the equation
Brown University Physics Department
Physics 0050/0070
K net =
1
Mv y (1 + I cm Mb 2 ) = Mgy 2
(9) 2 where y is the distance in the vertical direction, and differentiating with respect to time gives dv dy , or Mv y (1 + I cm Mb 2 ) = My dt dt Mv y a y (1 + I cm Mb 2 ) = Mgv y .
(10)
This is identical to equation 5 (after dividing bo th sides by v y). EXPERIMENT:
Experimentally you can measure the acceleration a and compare it with equation 5. If 1 you release the yoyo from rest then y = at 2 where y is the distance fallen and t is the 2 time of fall, t can be measured with a stop watch or a photogate timer. Videotaping the falling yoyo is also useful in obtaining position versus time data. The tension in the string of the falling yoyo can be measured with a spring force meter (available in lab) a nd can be compared with equation (6). Does your experimentally measured acceleration agree with theory? If you can experimentally determine the velocity of the yoyo near the bottom of its fall then you can directly check the validity of equation (9).
References : Resnick and Halliday p. 258-259 Problems 39 and 43 Ford p. 440 Problem 10.19 Acceleration of a pulled spool (L-16)