Practice taking derivatives of a position vector to calculate acceleration 1.10 * a particle moves in a circle (center o and radius r) with constant angular velocity ω counter- clockwise. the circle lies in the xy plane and the particle is on the x axis at time-0 show that the particle's position is given by r(t)r cos(wt) rsin(ot) find the particle's velocity and acceleration. what are the magnitude and direction of the acceleration? relate your results to well-known properties of uniform circular motion hint: you will want to use the fact that the vector r above points radially outward from the origin to the particle.]

Amass m = 74 kg slides on a frictionless track that has a drop, followed by a loop-the-loop with radius r = 18.4 m and finally a flat straight section at the same height as the center of the loop (18.4 m off the ground). since the mass would not make it around the loop if released from the height of the top of the loop (do you know why? ) it must be released above the top of the loop-the-loop height. (assume the mass never leaves the smooth track at any point on its path.) 1. what is the minimum speed the block must have at the top of the loop to make it around the loop-the-loop without leaving the track? 2. what height above the ground must the mass begin to make it around the loop-the-loop? 3. if the mass has just enough speed to make it around the loop without leaving the track, what will its speed be at the bottom of the loop? 4. if the mass has just enough speed to make it around the loop without leaving the track, what is its speed at the final flat level (18.4 m off the ground)? 5. now a spring with spring constant k = 15600 n/m is used on the final flat surface to stop the mass. how far does the spring compress?