We are given a disk that starts from rest and undergoes uniform angular acceleration \( \alpha \). We need to determine how the magnitude of the linear acceleration of a point on the rim of the disk changes with time \( t \).
### Step 1: Understanding Linear Acceleration
The total linear acceleration \( a \) of a point on the rim consists of two components:
1. **Tangential acceleration** \( a_t \), which is due to the angular acceleration \( \alpha \).
2. **Centripetal acceleration** \( a_c \), which arises due to the changing angular velocity.
#### Tangential Acceleration:
The tangential acceleration is given by:
\[ a_t = r \alpha \]
Since \( \alpha \) is constant, \( a_t \) does not depend on \( t \), meaning it remains constant.
#### Centripetal Acceleration:
Centripetal acceleration is given by:
\[ a_c = r \omega^2 \]
where \( \omega \) is the angular velocity. The angular velocity \( \omega \) for uniform angular acceleration is:
\[ \omega = \alpha t \]
Substituting \( \omega \) in the centripetal acceleration formula:
\[ a_c = r (\alpha t)^2 = r \alpha^2 t^2 \]
Thus, \( a_c \) grows as \( t^2 \).
### Step 2: Total Linear Acceleration
The total linear acceleration is given by:
\[ a = \sqrt{a_t^2 + a_c^2} \]
Substituting the values:
\[ a = \sqrt{(r \alpha)^2 + (r \alpha^2 t^2)^2} \]
\[ a = r \sqrt{\alpha^2 + \alpha^4 t^4} \]
### Step 3: Growth of \( a \) with Time
We analyze how \( a \) varies in two cases:
#### Case 1: Small \( t \) (\( \alpha t^2 \ll 1 \))
When \( t \) is small, \( \alpha^4 t^4 \) is much smaller than \( \alpha^2 \), so we approximate:
\[ a \approx r \alpha \]
Since this is independent of \( t \), \( a \) remains **constant**. None of the given options match this exactly, but if we were to choose, the closest option would be **t⁰ (constant), which is not listed**.
#### Case 2: Large \( t \) (\( \alpha t^2 \gg 1 \))
For large \( t \), \( \alpha^4 t^4 \) dominates, so we approximate:
\[ a \approx r \alpha^2 t^2 \]
Thus, \( a \) grows as \( t^2 \).
### Step 4: Answer Selection
- For small \( t \), \( a \) is constant. No option exactly matches, but if forced to choose, \( t^0 \) would be ideal.
- For large \( t \), \( a \propto t^2 \), so the correct choice is **(B) \( t^2 \)**.
Thus, the correct answer is:
For small \( t \), \( a \) remains constant (not listed).
For large \( t \), \( a \) grows as \( t^2 \) (**option B**).
