Question

Find the tangential and normal components of acceleration for a particle moving along the conical helix defined by \(r(t)=\left<t cost, t sin t, t \right>\).

Short answer

The tangential and normal components of acceleration for the position functions are \(a_{T}=\frac{t}{\sqrt{t^{2}+2}}\) and \(a_{N}=\sqrt{\frac{t^{4}+5t^{2}+8}{t^{2}+2}}\).

Step-by-step solution

Step 1. Find the tangential component of acceleration 

To find the tangential component of acceleration, we will use the formula \(a_{T}=\frac{v\cdot a}{\left\|v \right\|}\).

Now, if we differentiate \(r(t)\) we get \(r^{\prime}\left ( t \right )=v\left ( t \right )\) and \(r^{\prime \prime}\left ( t \right )=a\left ( t \right )\).

So,

\(r\left ( t \right )=\left<t cos t, t sin t, t \right>\)

\(v(t)=r^{\prime}\left ( t \right )=\left<-t sint +cost,t cos t +sin t,1 \right>\)

\(a(t)=r^{\prime \prime} \left ( t \right )=\left<-t cost -sint -sint, -t sin t +cos t + cos t,0 \right>\)

\(\left\|v \right\|=\left\| \left<-t sint +cost,t cos t +sin t,1 \right>\right\|\)

\(\left\| v\right\|=\sqrt{\left ( -t sint +cost \right )^{2}+\left ( t cos t +sin t \right )^{2}+1}\)

\(\left\| v\right\|=\sqrt{t^{2}\left ( sin^{2}t+cos^{2}t \right )+1+1}\)

\(\left\| v\right\|=\sqrt{t^{2}+2}\)

\(v\cdot a=v\left ( t \right )\cdot a\left ( t \right )\)

\(v\cdot a=\left<-t sint +cost,t cos t +sin t,1 \right> \cdot \left<-t cost -sint -sint, -t sin t +cos t + cos t,0 \right>\)

\(v\cdot a=2t-t\)

\(v\cdot a=t\)

Now, put all the above values we get in the formula

\(a_{T}=\frac{v\cdot a}{\left\|v \right\|}\)

\(a_{T}=\frac{t}{\sqrt{t^{2}+2}}\)

Step 2. Find the normal component of acceleration 

To find the normal component of acceleration, we will use the formula \(a_{N}=\frac{\left\|v\times a \right\|}{\left\|v \right\|}\).

So,

\(v\times a=v(t)\times a(t)\)

\(v\times a=\left|\begin{array}{ccc}i & j & k \\-t sint+cost & tcost+ sin t & 1 \\-tcost-2sint & -tsint+2cost & 0\end{array}\right|\)

\(v\times a=i(tsint-2cost)-j(tcost+2sint)+k(\left ( -t sint+cost \right )\left ( -tsint+2cost \right )-\left ( tcost+ sin t \right )\left ( -tcost-2sint \right ))\)

\(v\times a=i(tsint-2cost)-j(tcost+2sint)+k\left ( t^{2}+2 \right )\)

\(\left\|v\times a \right\|=\sqrt{\left ( tsint-2cost \right )^{2}+\left ( tcost+2sint \right )^{2}+\left ( t^{2}+2 \right )}\)

\(\left\|v\times a \right\|=\sqrt{t^{4}+5t^{2}+8}\)

\(\left\| v\right\|=\sqrt{t^{2}+2}\)

Now, put all the above values we get in the formula,

\(a_{N}=\frac{\left\|v\times a \right\|}{\left\|v \right\|}\)

\(a_{N}=\sqrt{\frac{t^{4}+5t^{2}+8}{t^{2}+2}}\)