Chapter 4: Q. 38 (page 107)

38. II FIGURE EX4.38 shows the angular-velocity-versus-time graph for a particle moving in a circle. How many revolutions does the object make during the first 4 s?

Expert verified

The number of revolution made by a particle moving in a circle during the first 4s is 9.55 rev.

Step by step solution

From the graph up to t =2s the particle is having angular acceleration and from time t = 4s the particle is having constant angular velocity.

Then for $t = 2 s$

$α_{2} = \frac{d ω}{d t} = \frac{20 rad / s - 0}{2 s} = 10 rad / s^{2}$

Initial angular velocity is

$ω_{0} = 0 rad / s$

$Then Δ θ_{2} = ω_{0} t + \frac{1}{2} α_{2} t^{2} = 0 + \frac{1}{2} (10 rad / s^{2}) {(2 s)}^{2} = 20 rad$

Now from $t = 2 s$ to $t = 4 s, ω = 20 rad / s$

Then

role="math" localid="1652331307170"

\begin{array}{l} Δ θ_{4} = ω Δ t \\ = (20 rad / s) 2 s \\ = 40 rad \end{array}

Then total angular displacement is

$\begin{array}{l} Δ θ = Δ θ_{L} + Δ θ_{4} \\ = 20 rad + 40 rad \\ = 60 rad \end{array}$

Now we know that $Δ θ = 2 π N$

Thus number of revolutions is

$\begin{array}{l} N = \frac{Δ θ}{2 π} \\ = \frac{60 rad}{2 π rad / rev} \\ = 9.55 rev \end{array}$

Therefore, the number of revolutions made by a particle moving in a circle during the first 4s is 9.55 rev.

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