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/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

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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