Course and Section_______
Names ___________________________
Date___________
_________________________________
ROTATIONAL DYNAMICS EXPERIMENT
Introduction:
In this experiment, you will study the motion of objects undergoing uniformly accelerated
rotational motion. Each object will be mounted so that it rotates about a fixed vertical axis. Its
rotational acceleration will be caused by the tension in a string wrapped around a spindle and
connected to a hanging weight, as shown. Using a Pasco Smart Pulley, you will measure the
linear acceleration of the hanging weight for two different configurations: A) disk mounted
horizontally as shown above, B) horizontal disk and a hoop.
Equipment: Rotational dynamics apparatus, mass, mass hanger, caliper, pulley, photogate
disk (M, R)
spindle (r)
m
Preliminary Questions:
1. In the figure above, a disk is shown on the platform. If the disk is replaced by a hoop (hollow
disk) of the same mass M and radius R, will the acceleration of the system increase, decrease, or
stay the same?
2. After the mass m is released, will its acceleration be less than, equal to, or greater than g ?
Will the acceleration depend on m ?
Theory
Draw a freebody diagram for the hanging weight and for the rotating mass system. For the
latter, it is important to indicate where forces are applied.
Newton’s 2nd law applied to the falling mass m yields
mg − T = ma , or T = m( g − a ) ,
(1)
where T is the tension in the string. Similarly, for the rotating system, if we ignore friction in the
spindle bearing,
Tr =
Ia
Ia
, or T = 2 ,
r
r
(2)
where I is the moment of inertia of the rotating system, r is the spindle radius, and Tr is the
torque applied by the string to the spindle. Combining Eq. (1) and (2) yields
Ia
.
r2
You will use the equation above to find I by making a linear fit using Excel.
m( g − a ) =
(3)
Theoretically,
2
A) I disk = 12 M disk Rdisk
B)
(4)
I total = I disk + I hoop , and
(
I hoop = 12 M hoop R12 + R22
)
(5)
Procedure
Connect the Smart Pulley to the Pasco interface box. Open Data Studio and select the Smart
Pulley setup. Open a graph so that you can plot velocity versus time using the Smart Pulley.
Measure the mass Mdisk and radius Rdisk of the solid disk, the mass Mhoop and inside and outside
radii (R1 and R2) of the hoop, and the radius r of the spindle (to be measured with the caliper).
Use hanging mass values (m) of 50 g, 100 g, 150 g, and 200 g.
Mdisk = __________
Mhoop = ______
Rdisk = _______
R1 = ______
R2 = ______
r = _________
A) Mount the disk horizontally. Hang a 50 g mass from the end of the string, allow it to fall
and cause the disk to rotate, and record the velocity as a function of time with Datastudio.
From the slope of your curve, calculate the acceleration. Repeat for the other three
values of m. Calculate also the quantity m(g-a).
A) Disk
m
a
m(g-a)
B) Place the hoop on top of the disk and repeat the measurements in A).
B) Disk + Hoop
m
a
m(g-a)
Analysis and questions
1. To find the momentum of inertia I, first find the slope of the equation m( g − a ) =
𝐼𝐼
Ia
.
r2
When it is written as y = (slope)·x then: y = m(g-a) and x = a and (slope) = 𝑟𝑟 2
To find the numerical value of the slope enter your data into Excel. Graph m(g-a) versus
a and do a linear fit to the data.
A) slope =
B) slope =
2. Using your measured value of r and the slope, calculate I and enter it in the table below
as Imeas.
3. Now calculate the theoretical values of I for each of the configurations using Eq. (4), (5)
and enter the values in the table below as Itheor.
Case A)
Itheor.=
Imeas.=
Case B)
Itheor.=
Imeas.=
4. Compare your ‘theoretical’ and ‘measured’ values of I. What factors might account for
the differences between these values?
A) %Error = | meas – theor | / | theor | x 100 = __________
B) %Error = | meas – theor | / | theor | x 100 = __________
5. Review your answers to the preliminary questions, and explain any changes you wish to
make to them after having done this experiment.