A yo-yo is made of 3 disks (same material) and the inner one is 3times smaller than the two outer ones (R/r=3). A string (small thickness) is wrapped around the center disk. What will its acceleration be equal to?

2 Answers
Aug 4, 2018

#a = 38/201 color(white)(l) g#

Explanation:

Let the mass of the center disk be #m#; each of the two outer disks has a lateral area of #3^2=9# times that of the center disk and would thus be #9 color(white)(l) m# in mass.

The yo-yo would experience a downward gravitational pull of magnitude #G = 19 color(white)(l) m * g# where #g# the gravitational acceleration.

https://www.ck12.org/physics/physics-of-a-yo-yo/lesson/Yo-Yo-Type-Problems-PPC/

Let #T# resembles the magnitude of the tension force the string applies on the yo-yo. Apply Newton's Second Law of Motion:

# G - T= Sigma F = 19 color(white)(l) m * a#

The three disk revolves around the center of the yo-yo. The equation

#I = 1/2 * m* r^2#

gives the moment of inertia for each of the disk. The yo-yo would, therefore, have a moment of inertia of

#Sigma I = 2 xx 1/2 * 9 color(white)(l) m * (3 color(white)(l) r)^2 + 1/2 * m * r^2 = 163/2 color(white)(l) m * r^2#

The tension force the string exerts on the yo-yo is applied at a distance #r# away from the center of rotation. Thus the torque:

#tau = T * r#

The string would drive the yo-yo to spin at an angular acceleration #alpha# of

#alpha = tau / (Sigma I) = (T * r)/(163//2 * m * r^2)#

The equation #a = alpha * r# relates the angular acceleration #alpha# to its linear counterpart #a#:

#a = alpha * r#
#color(white)(a) = r * tau / (Sigma I) #
#color(white)(a) = r * (T * r)/(163//2 * m * r^2)#
#color(white)(a) = (T)/(163//2 * m )#

Therefore

#T = 163/2 * m * a#

Substituting #T# back to the #Sigma F = m * a# expression and solve for #a#:

#19 color(white)(l) m * g - 163/2 * m * a = 19 color(white)(l) m * a#
#a = 38/201 color(white)(l) g#

Does this result make sense? For reference, a yo-yo consisting of a single disk would experience a liner acceleration equal to #2//3# that of the gravitational acceleration under such configurations. [1] The addition of the two disks- despite adding to the weigh of the yo-yo- increase its moment of inertia making it harder to spin.

Reference
[1] "Physics of a Yo-Yo", CK-12 Foundation, https://www.ck12.org/physics/physics-of-a-yo-yo/lesson/Yo-Yo-Type-Problems-PPC/

Aug 5, 2018

Based on earlier answer by @jacob-t-3

Explanation:

Let the mass of the central disk be #=m#.
Given is radius of outer disks #=R/r=3#.
The disks are made of same material. It is assumed that thickness of all three is same. Consequently, mass of each outer disk is proportional to its #"radius"^2#.

#=># Mass of each outer disk #=9\ m#
Total weight of yo-yo #M=(2xx9+1)\ m=19\ m#

chegg.com, edited not to scale
Total weight of yo-yo acts downwards #=Mg#

#:.# Downwards force acting on yo-yo#=19mg# .......(1)
where #g# is acceleration due to gravity.

Let #T# be the magnitude of the tension force in the string due to the yo-yo. Net downwards force

#F_"net"=Mg-T#

If #a# is acceleration produced in the yo-yo, from Newtons second Law of motion

#Mg-T=Ma# ......(2)

The three disks revolve around the center of the yo-yo. The moment of inertia of a disk rotating about its center of mass is given as

#I = 1/2 "mass"* "radius"^2#

Total moment of inertia of yo-yo is sum of moments inertia of three disks.

#Sigma I = 2 (1/2 * 9 m * (3 r)^2) + 1/2 m * r^2 #
#=>Sigma I = 162/2\ mcdot r^2 + 1/2 * m * r^2#
#=>Sigma I = 163/2 \ m * r^2#

Weight, which is acting at the center of mass, donot produce any torque. Therefore, total torque #tau# produced is by tension #T# only.

#tau = T * r#

Let this torque produce an angular acceleration #alpha# in the yo-yo

#:.alpha = tau / (Sigma I) = (T * r)/(163/2\ m * r^2)# ....(3)

Expression relating the angular acceleration #alpha# to its acceleration #a# is

#a = r*alpha #

Using (3) we get

#a = r * (T * r)/(163/2 * m * r^2)#
#=>T = 163/2 * m * a#

Substituting #T# in (2), using (1) and solving for #a# we get

#19\ m * g - 163/2 \ m * a = 19\ m * a#
#=>19\ m * g= 163/2 \ m * a + 19\ m * a#
#=>19\ m * g= 201/2 \ m * a#
#=>a = 38/201\ g#