Question
Download Solution PDFA rod of length L fixed at the origin at one end is placed along the x-axis at t = 0. It starts rotating with a constant angular acceleration α about the z-axis in the x-y plane in the anti-clockwise as seen from the positive z-axis. The time taken to complete the first revolution is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCONCEPT:
- Kinematic equations of rotational motion about a fixed axis,
\(\Rightarrow ω = \frac {d θ}{dt}\)
\(\Rightarrow α= \frac {d ω}{dt}\)
\(\Rightarrow α= ω \frac {d ω}{d θ}\)
where ω is the instantaneous angular velocity, α is the instantaneous angular acceleration, θ is the angular displacement, and t is time.
- Kinematic equations of rotational motion with constant angular acceleration about a fixed axis,
\(\Rightarrow ω = ω _0 + α t\)
\(\Rightarrow θ = ω _0 t + \frac {1}{2} α t^2\)
\(\Rightarrow ω ^2 = ω _0 ^2 + 2 α θ\)
where ω0 is the initial angular velocity, ω is the final angular velocity, α is the angular acceleration, θ is the angular displacement, and t is time.
EXPLANATION:
Given: Length of the rod = L, and angular acceleration = α.
Using the second equation,
\(\Rightarrow θ = ω _0 t + \frac {1}{2} α t^2\)
\(\Rightarrow 2 \pi = 0 \times t + \frac {1}{2} α t^2\)
\(\Rightarrow T = \sqrt \frac {4\pi}{\alpha}\)
- Therefore option 4 is correct.
Last updated on Jul 4, 2025
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