91影视

Mass of the particle m=1.7 x 10^-27 kg

Circular trajectory = $\stackrel{鈫�}{r}=c\cos \left(k{t}^2\right)\widehat{谋}+c\sin \left(k{t}^2\right)\widehat{确}$

c= 5 m

k= 8 x 10⁴rad/sec²

We can write an equation of particle motion is given as,

$\stackrel{鈫�}{r}=c\cos \left(k{t}^2\right)\widehat{谋}+c\sin \left(k{t}^2\right)\widehat{确}$

Where,
c =radius of the circular path

So we can say radius of the path is 5 m

Mass of the particle m=1.7 x 10^-27 kg

Circular trajectory $\stackrel{鈫�}{r}=c\cos \left(k{t}^2\right)\widehat{谋}+c\sin \left(k{t}^2\right)\widehat{确}$

c= 5 m

k= 8 x 10⁴rad/sec²

Velocity is calculated as

$$\begin{gathered} \stackrel{鈫�}{v}=\frac{d\stackrel{鈫�}{r}}{dt}鈭� \\ =\frac{d}{dt}\left(c\cos \left(k{t}^2\right)\hat{i}+c\sin \left(k{t}^2\right)\hat{j}\right) \\ =2kct\left(-\sin \left(k{t}^2\right)\hat{i}+\cos \left(k{t}^2\right)\hat{j}\right) \end{gathered}$$

Substitute the value we get

$$\begin{gathered} v=2脳(8脳{10}^{4}rad/s^2)脳\left(5m\right)脳\left(3s\right)[-sin(8脳{10}^{4}rad/s^2)脳{\left(3s\right)}^2)i+cos(8脳{10}^{4}rad/s^2)脳{\left(3s\right)}^2\left)j\right] \\ v=2.4脳{10}^{6}m/s \end{gathered}$$

Part(c) Step 1 : given information

Mass of the particle m=1.7 x 10^-27 kg

Circular trajectory = $\stackrel{鈫�}{r}=c\cos \left(k{t}^2\right)\widehat{谋}+c\sin \left(k{t}^2\right)\widehat{确}$

c= 5 m

k= 8 x 10⁴rad/sec²

Force is calculated as

$F鈨�=ma鈨�$

Lets find acceleration

$$\begin{gathered} \stackrel{鈫�}{a}=\stackrel{鈫�}{v}=\frac{d\stackrel{鈫�}{r}}{dt}=\frac{d\stackrel{鈫�}{v}}{dt}=\frac{d}{dt}\left(2kct\left(-\sin \left(k{t}^2\right)\hat{i}+\cos \left(k{t}^2\right)\hat{j}\right)\right) \\ =2kc\frac{d}{dt}\left(t\left(-\sin \left(k{t}^2\right)\hat{i}+\cos \left(k{t}^2\right)\hat{j}\right)\right) \\ =2kc\left[\left(-\sin \left(k{t}^2\right)\hat{i}+\cos \left(k{t}^2\right)\hat{j}\right)+2k{t}^2\left(-\cos \left(k{t}^2\right)\hat{i}-\sin \left(k{t}^2\right)\hat{j}\right)\right] \\ k{t}^2=8脳{10}^4脳3^2=7.2脳{10}^5 \\ \sin \left(k{t}^2\right)=-0.36\text{and}\cos \left(k{t}^2\right)=-0.93 \end{gathered}$$

$$\begin{gathered} \stackrel{鈫�}{a}=2脳(8脳{10}^{4}rad/se{c}^2)脳\left(5m\right)\left[(-0.36\hat{i}-0.93\hat{j})+2脳7.2脳{10}^5(0.93\hat{i}+0.36\hat{j})\right] \\ =8脳{10}^5脳2脳7.2脳{10}^5(0.93\hat{i}+0.36\hat{j}) \\ =1.15脳{10}^{12}(0.93\hat{i}+0.36\hat{j})m/s^2 \end{gathered}$$

Now substitute values we get

$$\begin{gathered} \stackrel{鈫�}{F}=m\stackrel{鈫�}{a}=(1.7脳{10}^{-27}kg)脳(1.15脳{10}^{12}(0.93\hat{i}+0.36\hat{j})m/se{c}^2) \\ =1.96脳{10}^{-15}(0.93\hat{i}+0.36\hat{j})N \\ =\left(1.82脳{10}^{-15}\hat{i}+0.71脳{10}^{-15}\hat{j}\right)\mathrm{N} \end{gathered}$$