Question

23. FIGURE EX4.23 shows the angular-velocity-versus-time graph for a particle moving in a circle. How many revolutions does the object make during the first 4s

Short answer

The number if revolution made by the object during first 4s is 10.34rev.

Step-by-step solution

. Given information

The graph is drawn between angular velocity versus time for a particle.

The following graph shows the graph of angular velocity versus time of a particle for first 4s.

Step 2. Explanation

The formula to calculate the area of ABC is,

$A_{OABC}=OA\times AB$

The formula to calculate the area of BCE is,

$A_{BCE}=\frac{1}{2}\left(CE\right)\left(BC\right)$

The formula to calculate the area of DEF is,

$A_{DEF}=\frac{1}{2}\left(DF\right)\left(EF\right)$

The angular displacement of the particle in first from the shown figure (I) is,

$\begin{array}{l}\theta =A_{OABC}+A_{BCE}+A_{DEF}\\ \theta =(OA\times AB)+\left(\frac{1}{2}\left(CE\right)\left(BC\right)\right)+\left(\frac{1}{2}\left(DF\right)\left(EF\right)\right)\\ \theta =\left(\right(20\text{rad}/\text{s})\times (2\text{s}\left)\right)+\left(\frac{1}{2}(2\text{s})(20\text{rad}/\text{s})\right)+\left(\frac{1}{2}(1\text{s})(10\text{rad}/\text{s})\right)\\ =40\text{rad}+20\text{rad}+5\text{rad}\\ =\left(65\text{rad}\times \frac{1\text{rev}}{2\pi \text{rad}}\right)\\ =10.34\text{rev}\end{array}$