Question Number 215410 by York12 last updated on 05/Jan/25
$$\mathrm{The}\:\mathrm{following}\:\mathrm{diagram}\:\mathrm{shows}\:\mathrm{the}\:\mathrm{relationship} \\ $$$$\mathrm{between}\:\mathrm{electromotive}\:\mathrm{force}\:\left(\mathrm{e}.\mathrm{m}.\mathrm{f}\right)\:\mathrm{and}\:\mathrm{the}\:\mathrm{time}\:\left(\mathrm{t}\right)\:\mathrm{in}\:\mathrm{a}\:\mathrm{dynamo} \\ $$$$\mathrm{coil}.\mathrm{During}\:\mathrm{the}\:\mathrm{time}\:\mathrm{interval}\:\mathrm{from}\:\mathrm{t}=\mathrm{0}\:\mathrm{to}\:\mathrm{t}=\frac{\mathrm{1}}{\mathrm{30}}\:\mathrm{seconds},\: \\ $$$$\mathrm{the}\:\mathrm{average}\:\mathrm{electromotive}\:\mathrm{force}\:\left(\mathrm{e}.\mathrm{m}.\mathrm{f}\right)\:\mathrm{induced}\:\mathrm{in}\:\mathrm{the}\:\mathrm{coil}\:\mathrm{is}: \\ $$$$\left(\mathrm{a}\right)\mathrm{42}.\mathrm{46}\:{V} \\ $$$$\left(\mathrm{b}\right)\mathrm{19}.\mathrm{11}\:{V} \\ $$$$\left(\mathrm{c}\right)\mathrm{127}.\mathrm{39}\:{V} \\ $$$$\left(\mathrm{d}\right)\mathrm{173}.\mathrm{21}\:{V} \\ $$
Commented by York12 last updated on 05/Jan/25
Commented by York12 last updated on 05/Jan/25
![My idea was emf=ABNwsin(θ)=ABN((d(θ))/(d(t))) sin (θ) ⇒emf d(t)= ABNsin (θ)d(θ) ⇒∫_t_1 ^t_2 emf d(t)=ABN∫_θ_1 ^θ_2 sin (θ)d(θ)=((emf_(max) )/w)∫_θ_1 ^θ_2 sin (θ)d(θ) =((emf_(max) )/w)[cos (θ_1 )−cos (θ_2 )]=((emf_(max) )/w)[cos (wt_1 )−cos (wt_2 )] I do not know what is wrong with my approach , please help.](https://www.tinkutara.com/question/Q215412.png)
$$\mathrm{My}\:\mathrm{idea}\:\mathrm{was} \\ $$$$\mathrm{emf}=\mathrm{A}{BNw}\mathrm{sin}\left(\theta\right)={ABN}\frac{{d}\left(\theta\right)}{{d}\left({t}\right)}\:\mathrm{sin}\:\left(\theta\right) \\ $$$$\Rightarrow{emf}\:{d}\left({t}\right)=\:{ABN}\mathrm{sin}\:\left(\theta\right){d}\left(\theta\right) \\ $$$$\Rightarrow\underset{{t}_{\mathrm{1}} } {\overset{{t}_{\mathrm{2}} } {\int}}{emf}\:{d}\left({t}\right)={ABN}\underset{\theta_{\mathrm{1}} } {\overset{\theta_{\mathrm{2}} } {\int}}\mathrm{sin}\:\left(\theta\right){d}\left(\theta\right)=\frac{{emf}_{{max}} }{{w}}\underset{\theta_{\mathrm{1}} } {\overset{\theta_{\mathrm{2}} } {\int}}\mathrm{sin}\:\left(\theta\right){d}\left(\theta\right) \\ $$$$=\frac{{emf}_{{max}} }{{w}}\left[\mathrm{cos}\:\left(\theta_{\mathrm{1}} \right)−\mathrm{cos}\:\left(\theta_{\mathrm{2}} \right)\right]=\frac{{emf}_{{max}} }{{w}}\left[\mathrm{cos}\:\left({wt}_{\mathrm{1}} \right)−\mathrm{cos}\:\left({wt}_{\mathrm{2}} \right)\right] \\ $$$$\mathrm{I}\:\mathrm{do}\:\mathrm{not}\:\mathrm{know}\:\mathrm{what}\:\mathrm{is}\:\mathrm{wrong}\:\mathrm{with}\:\mathrm{my}\:\mathrm{approach}\:,\: \\ $$$$\mathrm{please}\:\mathrm{help}. \\ $$$$ \\ $$