The correct answer is B because:

List the known quantities:
- Rate of rotation = 3 revolutions per minute
- Angular displacement, Δθ = 360π radians
- Radius, r = 25 cm = 0.25 m
- Linear acceleration, a = 1.75 × 10⁻⁴ m s⁻²
Select the correct angular acceleration equation from the data booklet:
- $∆ θ = ω_{i} t + \frac{1}{2} α t$
First, determine the initial angular velocity, ω_i:
- 3 revolutions per minute = 3 × 2π radians moved in 60 seconds
- $ω_{i} = \frac{6 π}{60}$
- $ω_{i} = \frac{π}{10}$
Second, calculate the angular acceleration of the ballerina, α:
- $α = \frac{a}{r}$
- $α = \frac{(1.75 \times 10^{- 4})}{0.25}$
- $α = 7 \times 10^{- 4}$ rad s⁻²
Now substitute the known quantities and rearrange the quadratic to find the time of the performance, t:
- $∆ θ = ω_{i} t + \frac{1}{2} α t$
- $360 π = \frac{π}{10} t + (\frac{1}{2} \times (7 \times 10^{- 4}) t^{2})$
- $(3.5 \times 10^{- 4}) t^{2} + \frac{π}{10} t - 360 π = 0$
- $t = \frac{- \frac{π}{10} \pm \sqrt{{(\frac{π}{10})}^{2} - [4 \times (3.5 \times 10^{- 4}) \times - 360 π]}}{(2 \times 3.5 \times 10^{- 4})}$
- $t = 1404 s$
Finally, convert the time t from seconds into minutes:
- 1404 s = 1404 ÷ 60 = 23.4 minutes
- To the nearest whole minute t = 23 minutes

Rigid Body Mechanics (HL IB Physics)

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