give answer in 2 step with explanation at the end of each step and final answer at the end:10f2
Rotational Dynamics
Purpose: Tis bovis he reslanip been the apple org na oan disk and
Ine eng guar ev

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:10f2
Rotational Dynamics
Purpose: Tis bovis he reslanip been the apple org na oan disk and
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bop deeb Tindal Loser py ids spr
Sma cll te mnt of het. nh experimen, wil i enon. We wil
Seth velo of 31 Rewher oe vf he ik ad ts rods
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Ia dok Th sin hu over (sls. coe) ly he cd of he le and
Vl banger 11s connate oh Val ates (me) ty be add tte ange The
obo it beers i 1 tied bl ae oda on tt Sig
Vhich nr produces argu och Potzonal dk
ear sce fhe roping dk whe coped by messing the nes seclertion
Oe dceng mts, Ti ik csmplabed in he ere Pro oan sera andy
Somme a sepa ih loi Bap cuted ho i seeming as The sion brgeoe
oe tar colton of dein he reso eat fe Ak £m
Where 1s pin he us of he male dk.
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The hearst squtions are
oon 1 gTemua where ms he hanging mas, gs he acceleration du 10
ry 1 he scrum he hanging miss
Couation: Pima where ish smal ai sh a1 te sope fom the veocy graph
whe dsc mes
Bouto

VIPSolutions · Accepted Answer

## Step 1: Relate Torque and Angular Acceleration

**Explanation:**  
From the information and equations given:
- **Equation 3:** $ \tau = rT $ (torque is tension times radius)
- **Equation 4:** $ \tau = I\alpha $ (Newton's 2nd law for rotation)
  
Set these equal to each other:
$$
rT = I\alpha
$$
From **Equation 2:** $ \alpha = \frac{a}{r} $ (angular acceleration relates to linear acceleration)

Substitute this into the torque equation:
$$
rT = I\left(\frac{a}{r}\right)
$$
$$
rT = \frac{I a}{r}
$$
$$
T = \frac{I a}{r^2}
$$

---

## Step 2: Relate Tension to Linear Acceleration of Hanging Mass

**Explanation:**  
From **Equation 1:** $ m_2 g - T = m_2 a $

Solve for $ T $:
$$
T = m_2 g - m_2 a
$$

Now set this equal to the expression for $ T $ from Step 1:
$$
m_2 g - m_2 a = \frac{I a}{r^2}
$$

Bring like terms together:
$$
m_2 g = m_2 a + \frac{I a}{r^2}
$$
$$
m_2 g = a \left( m_2 + \frac{I}{r^2} \right)
$$

Solve for $ a $ (theoretical acceleration):
$$
a_{\text{theoretical}} = \frac{m_2 g}{ m_2 + \frac{I}{r^2} }
$$

---

## **Final Answer**

$$
\boxed{
a_{\text{theoretical}} = \frac{m_2 g}{ m_2 + \frac{I}{r^2} }
}
$$

Where:
- $ m_2 $ = hanging mass  
- $ g $ = acceleration due to gravity  
- $ I $ = moment of inertia of the disk ($ I = \frac{1}{2} m_1 R^2 $ for a solid disk)  
- $ r $ = small radius where string is wound

VIPSolutions · Answer

# Rotational Dynamics Problem

## Step 1: Relate Torque and Angular Acceleration

**Explanation:**  
From the information and equations given:
- **Equation 3:** $ \tau = rT $ (Torque is the product of tension and radius)
- **Equation 4:** $ \tau = I\alpha $ (Newton's second law for rotation)

Setting these equations equal to each other gives:
$$
rT = I\alpha
$$
Using **Equation 2:** $ \alpha = \frac{a}{r} $ (Angular acceleration is related to linear acceleration), we can substitute:
$$
rT = I\left(\frac{a}{r}\right)
$$
This simplifies to:
$$
rT = \frac{I a}{r}
$$
Rearranging yields:
$$
T = \frac{I a}{r^2}
$$

---

## Step 2: Relate Tension to Linear Acceleration of Hanging Mass

**Explanation:**  
From **Equation 1:** $ m_2 g - T = m_2 a $ (The net force on the hanging mass), we can solve for $ T $:
$$
T = m_2 g - m_2 a
$$

Now, equate this expression for $ T $ with the one derived in Step 1:
$$
m_2 g - m_2 a = \frac{I a}{r^2}
$$

Bringing like terms together gives:
$$
m_2 g = m_2 a + \frac{I a}{r^2}
$$
Factoring out $ a $:
$$
m_2 g = a \left( m_2 + \frac{I}{r^2} \right)
$$

To find $ a $ (theoretical acceleration), we solve for $ a $:
$$
a_{\text{theoretical}} = \frac{m_2 g}{ m_2 + \frac{I}{r^2} }
$$

---

## **Final Answer**

$$
\boxed{
a_{\text{theoretical}} = \frac{m_2 g}{ m_2 + \frac{I}{r^2} }
}
$$

### Where:
- $ m_2 $ = hanging mass
- $ g $ = acceleration due to gravity
- $ I $ = moment of inertia of the disk (for a solid disk, $ I = \frac{1}{2} m_1 R^2 $)
- $ r $ = radius where the string is wound around the disk

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