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In class he said to picture the wheel being stationary. So from my understanding with the wheel being in the xz plane it is rotating around the axis (0,0). Picture a bicycle lifted for maintenance, the front tire spins freely without actually rolling anywhere. The bug is crawling outwards(from (0,0) towards edge of rim) on the spoke, while the spoke is spinning with the wheel at 12 radians per second. So from what I understand, we have to take the bugs motion, which is linear, and factor in the fact that the spoke he is crawling on is also moving, but in a circular motion. So the bugs movement should be something like p(t) =$\langle0,3\rangle t$ (supposing the spoke lies along the positive z axis) and the circular motion being p(t) =$ \langle \cos(t),\sin(t)\rangle $,(since the wheel is a circle) now since the wheel is spinning at 12 radians per second, this is about 1.9 revolutions per second $ \frac {12radians}{second}$ * $ \frac{1revolution}{2\pi radians}$=1.9$\frac{revolution}{second}$ . So this now makes your circular motion p(t) = $\langle \cos(1.9t),\sin(1.9t)\rangle$ . Now as far as him saying we have to do some multiplication to get the final vector function I'm not sure. I don't know if this will help you at all, but hopefully someone else can help elaborate further from this point, or clarify further for the two of us.

In class he said to picture the wheel being stationary. So from my understanding with the wheel being in the xz plane it is rotating around the axis with it's center being (0,0). Picture a bicycle lifted for maintenance, the front tire spins freely without actually rolling anywhere. The bug is crawling outwards(from (0,0) towards edge of rim) on the spoke, while the spoke is spinning with the wheel at 12 radians per second. So from what I understand, we have to take the bugs motion, which is linear, and factor in the fact that the spoke he is crawling on is also moving, but in a circular motion. So the bugs movement should be something like p(t) =$\langle0,3\rangle t$ (supposing the spoke lies along the positive z axis) and the circular motion being p(t) =$ \langle \cos(t),\sin(t)\rangle $,(since the wheel is a circle) now since the wheel is spinning at 12 radians per second, this is about 1.9 revolutions per second $ \frac {12radians}{second}$ * $ \frac{1revolution}{2\pi radians}$=1.9$\frac{revolution}{second}$ . So this now makes your circular motion p(t) = $\langle \cos(1.9t),\sin(1.9t)\rangle$ . Now as far as him saying we have to do some multiplication to get the final vector function I'm not sure. I don't know if this will help you at all, but hopefully someone else can help elaborate further from this point, or clarify further for the two of us.

In class he said to picture the wheel being stationary. So from my understanding with the wheel being in the xz plane it is rotating with it's center being (0,0). Picture a bicycle lifted for maintenance, the front tire spins freely without actually rolling anywhere. The bug is crawling outwards(from (0,0) towards edge of rim) on the spoke, while the spoke is spinning with the wheel at 12 radians per second. So from what I understand, we have to take the bugs motion, which is linear, and factor in the fact that the spoke he is crawling on is also moving, but in a circular motion. So the bugs movement should be something like p(t) =$\langle0,3\rangle t$ (supposing the spoke lies along the positive z axis) and the circular motion being p(t) =$ \langle \cos(t),\sin(t)\rangle $,(since the wheel is a circle) now since the wheel is spinning at 12 radians per second, this is about 1.9 revolutions per second $ \frac {12radians}{second}$ * $ \frac{1revolution}{2\pi radians}$=1.9$\frac{revolution}{second}$ . So this now makes your circular motion p(t) = $\langle \cos(1.9t),\sin(1.9t)\rangle$ . Now as far as him saying we have to do some multiplication to get the final vector function I'm not sure. I don't know if this will help you at all, but hopefully someone else can help elaborate further from this point, or clarify further for the two of us.

In class he said to picture the wheel being stationary. So from my understanding with the wheel being in the xz plane it is rotating with it's center being (0,0). Picture a bicycle lifted for maintenance, the front tire spins freely without actually rolling anywhere. The bug is crawling outwards(from (0,0) towards edge of rim) on the spoke, while the spoke is spinning with the wheel at 12 radians per second. So from what I understand, we have to take the bugs motion, which is linear, and factor in the fact that the spoke he is crawling on is also moving, but in a circular motion. So the bugs movement should be something like p(t) =$\langle0,3\rangle t$ (supposing the spoke lies along the positive z axis) and the circular motion being p(t) =$ \langle \cos(t),\sin(t)\rangle $,(since the wheel is a circle) now since the wheel is spinning at 12 radians per second, this is about 1.9 revolutions per second $ \frac {12radians}{second}$ * $ \frac{1revolution}{2\pi radians}$=1.9$\frac{revolution}{second}$ . So this now makes your circular motion p(t) = $\langle \cos(1.9t),\sin(1.9t)\rangle$ . Now as far as him saying we have to do some multiplication to get the final vector function I'm not sure. I don't know if this will help you at all, but hopefully someone else can help elaborate further from this point, or clarify further for the two of us.

added on later I'm not sure if this is even close to being right, but when I thought more about him saying we needed to multiply the two, I ended up coming up with p(t)= $\langle3t(cos(1.9t)),3t(sin(1.9t))\rangle$

Since the bugs motion is 3 units per second, it would make sense that for every t he would move 3 units, so multiplying that into the circular motion of the wheel is how I ended up with the above equation.

In class he said to picture the wheel being stationary. So from my understanding with the wheel being in the xz plane it is rotating with it's center being (0,0). (0,0,0). Picture a bicycle lifted for maintenance, the front tire spins freely without actually rolling anywhere. The bug is crawling outwards(from (0,0) (0,0,0) towards edge of rim) on the spoke, while the spoke is spinning with the wheel at 12 radians per second. So from what I understand, we have to take the bugs motion, which is linear, and factor in the fact that the spoke he is crawling on is also moving, but in a circular motion. So the bugs movement should be something like p(t) =$\langle0,3\rangle =$\langle0,0,3\rangle t$ (supposing the spoke lies along the positive z axis) and the circular motion being p(t) =$ \langle \cos(t),\sin(t)\rangle \cos(t),0,\sin(t)\rangle $,(since the wheel is a circle) now since the wheel is spinning at 12 radians per second, this is about 1.9 revolutions per second $ \frac {12radians}{second}$ * $ \frac{1revolution}{2\pi radians}$=1.9$\frac{revolution}{second}$ . So this now makes your circular motion p(t) = $\langle \cos(1.9t),\sin(1.9t)\rangle$ \cos(1.9t),0,\sin(1.9t)\rangle$ . Now as far as him saying we have to do some multiplication to get the final vector function I'm not sure. I don't know if this will help you at all, but hopefully someone else can help elaborate further from this point, or clarify further for the two of us.

added on later I'm not sure if this is even close to being right, but when I thought more about him saying we needed to multiply the two, I ended up coming up with p(t)= $\langle3t(cos(1.9t)),3t(sin(1.9t))\rangle$ $\langle3t(cos(1.9t)),0,3t(sin(1.9t))\rangle$

Since the bugs motion is 3 units per second, it would make sense that for every t he would move 3 units, so multiplying that into the circular motion of the wheel is how I ended up with the above equation.

In class he said to picture the wheel being stationary. So from my understanding with the wheel being in the xz plane it is rotating with it's center being (0,0,0). Picture a bicycle lifted for maintenance, the front tire spins freely without actually rolling anywhere. The bug is crawling outwards(from (0,0,0) towards edge of rim) on the spoke, while the spoke is spinning with the wheel at 12 radians per second. So from what I understand, we have to take the bugs motion, which is linear, and factor in the fact that the spoke he is crawling on is also moving, but in a circular motion. So the bugs movement should be something like p(t) =$\langle0,0,3\rangle t$ (supposing the spoke lies along the positive z axis) and the circular motion being p(t) =$ \langle \cos(t),0,\sin(t)\rangle $,(since the wheel is a circle) now since the wheel is spinning at 12 radians per second, this is about 1.9 revolutions per second $ \frac {12radians}{second}$ * $ \frac{1revolution}{2\pi radians}$=1.9$\frac{revolution}{second}$ . So this now makes your circular motion p(t) = $\langle \cos(1.9t),0,\sin(1.9t)\rangle$ . Now as far as him saying we have to do some multiplication to get the final vector function I'm not sure. I don't know if this will help you at all, but hopefully someone else can help elaborate further from this point, or clarify further for the two of us.

added on later I'm not sure if this is even close to being right, but when I thought more about him saying we needed to multiply the two, I ended up coming up with p(t)= $\langle3t(cos(1.9t)),0,3t(sin(1.9t))\rangle$

Since the bugs motion is 3 units per second, it would make sense that for every t he would move 3 units, so multiplying that into the circular motion of the wheel is how I ended up with the above equation.