A force acts on a tyre, causing it to rotate about its centre. The tyre has a constant angular acceleration of 26.4 rad s–2 and was initially at rest.(a) Deduce the angular displacement θ of the tyre after 12 full rotations.
Question
A force acts on a tyre, causing it to rotate about its centre. The tyre has a constant angular acceleration of 26.4 rad s–2 and was initially at rest.(a) Deduce the angular displacement θ of the tyre after 12 full rotations.
Solution
To solve this problem, we first need to understand what a full rotation means in terms of radians. One full rotation is equal to 2π radians. Therefore, 12 full rotations would be 12 * 2π = 24π radians.
However, the problem is asking for the angular displacement after the tyre has undergone an angular acceleration, not just rotations. The angular displacement (θ) can be found using the equation of motion:
θ = ω₀t + 0.5αt²
where: ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time.
Given that the tyre was initially at rest, ω₀ = 0. Also, we know that α = 26.4 rad/s². However, we don't have the time (t) it took for the tyre to make 12 full rotations.
Unfortunately, without the time, we cannot calculate the angular displacement due to acceleration. We can only say that the tyre rotated 24π radians due to the 12 full rotations.
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