Can someone help with two problems?

Question

Trick is to complete a vertical loop on a mortorcycle. this trick is dangerous because if the motorcycle does not travel with enough speed, the rider falls off the track before reaching the top of the loop. what is the minimum speed necessary for a rider to sucessfully go around in a vertical loop of 10 meters?

huge mass spring system is undergoing simple harmoic motion with angular frequency and amplitude. find its speed at the point where the kinetic and potential energies are equal.


Thanks!

Answer 1

help with the first one. Lucky for you I found a similar problem in a physics book, that of a bicycle doing a loop where the height of the loop is 5.4 meters.

The loop will exert a Normal force on the bike (N) in the vertical direction, specifically straight down. Also the weight of the bike and rider (W) is a force acting straight down. W=mg

Acceleration (centripetal acceleration) also is pointing towards the center of the loop, given as a=v^2/R

You should draw a free-body diagram to show those factors.

Then sum the forces:

-N - mg = -ma (Newton's second law)

-N - mg = -mv^2/R

but at the top of the loop, we don't want the rider to fall, but the minimum speed he can attain is that when the loop no longer exerts a normal force on him, so N=0

-mg = -mv^2/R

after some algebra:

g = v^2/R

You know g (9.8 m/s2), you know R = 5 meters (half of loop)

So the minimum velocity is v = sqrt(gR)

v=sqrt(9.8*5)

v=7 m/s

Hope you get the 2nd one...and I hope this helped, without you just copying stuff...try and understand it, or the test will be painful!

Answer 2

I'll take on the second problem since there are nice answers to the first one. The oscillation is a sine wave.. At the extreme positions, all the energy is potential, as the motion is briefly zero while the oscillation is reversing. When crossing the axis, all the energy is kinetic: there is no restoring force. The place where the potential energy and kinetic energy will be equal is 0.5 of the maximum potential or kinetic energy, or sqrt(0.5) of the maximum velocity.

Answer 3

45mph.

When the spring is flat.