Question

Two metal disks, one with radius R1 = 2.45 cm and mass M1 = 0.900 kg and the other with radius R2 = 5.00 cm and mass M2 = 1.60 kg, are welded together and mounted on a frictionless axis through their common center. A light string is wrapped around the edge of the smaller disk and a 1.60 kg block is suspended from the free end of the string. (a) What is the magnitude of the downward acceleration of the block after it is released? (b) Repeat the calculation of part (a), this time with the string wrapped around the edge of the larger disk.

Answer 1

part (a)
`a_1\ =\ 2.9\ kg`

Part (b)
`a_2\ =\ 6.25\ kg`

Step-by-step explanation:

Given,

• mass of the smaller disk =
  `M_1\ =\ 0.900\ kg`
• Radius of the smaller disk =
  `R_1\ =\ 2.45\ cm\ =\ 0.0245\ m`

• mass of the larger disk =
  `M_2\ =\ 1.6\ kg`
• Radius of the larger disk =
  `R_2\ =\ 5.0\ cm\ =\ 0.05\ m`
• mass of the hanging block = m = 1.60 kg

Let I be the moment of inertia of the both disk after the welding,
`\therefore I\ =\ I_1\ +\ I_2\\\Rightarrow I\ =\ (1)/(2)(M_1R_1^2\ +\ M_2R_2^2)\\\Rightarrow I\ =\ 0.5* (0.9* 0.0245^2\ +\ 1.6* 0.05^2)\\\Rightarrow I\ =\ 2.27* 10^(-3)\ kgm^2`

part (a)

A block of mass m is hanging on the smaller disk,

From the f.b.d. of the block,

Let 'a' be the acceleration of the block and 'T' be the tension in the string.

`mg\ -\ T\ =\ mg\\\Rightarrow T\ =\ mg\ -\ ma\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,eqn (1)`

Net torque on the smaller disk,

`\therefore \tau\ =\ I\alpha\\\Rightarrow TR_1\ =\ (Ia)/(R_1)\\\Rightarrow T\ =\ (Ia)/(R_1^2)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,enq (2)`

From eqn (1) and (2), we get,

`mg\ -\ ma\ =\ (Ia)/(R_1^2)\\\Rightarrow a\ =\ (mg)/((I)/(R_1^2)\ +\ m)\\\Rightarrow a\ =\ (1.60* 9.81)/((2.27* 10^(-3))/(0.027^2)\ +\ 1.60)\\\Rightarrow a\ =\ 2.91\ m/s^2`

part (b)

In this case the mass is rapped on the larger disk,

From the above expression of the acceleration of the block, acceleration is only depended on the radius of the rotating disk,

Let '
`a_2`' be the acceleration of the block in the second case,

From the above expression,

`\therefore a\ =\ (mg)/((I)/(R_1^2)\ +\ m)\\\Rightarrow a\ =\ (1.60* 9.81)/((2.27* 10^(-3))/(0.05^2)\ +\ 1.60)\\\Rightarrow a\ =\ 6.25\ m/s^2`