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Kinematics

Problem 1.56

A solid body rotates with angular velocity \(\omega=a t \mathbf{i}+b t^{2} \mathbf{j},\) where \(a=0.50 \mathrm{rad} / \mathrm{s}^{2}, b=0.060 \mathrm{rad} / \mathrm{s}^{3},\) and \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors of the \(x\) and \(y\) axes. Find: (a) the moduli of the angular velocity and the angular acceleration at the moment \(t=10.0 \mathrm{~s}\); (b) the angle between the vectors of the angular velocity and the angular acceleration at that moment.

Reveal Answer
 (a) ω=at1+(bt/a)2=8rad/s,β=a1+(2bt/a)2==1.3rad/s2; (b) 17∘\text { (a) } \omega=a t \sqrt{1+(b t / a)^{2}}=8 \mathrm{rad} / \mathrm{s}, \beta=a \sqrt{1+(2 b t / a)^{2}}==1.3 \mathrm{rad} / \mathrm{s}^{2} ; \text { (b) } 17^{\circ}
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