A cart at an amusement ride moves through a vertical circle with a radius of 10m. If the car initially has a speed of 22.18m/s A) what is the final speed at the top of the loop? B) What is the difference in the normal force on a person of 60kg at the top and at the bottom of the loop (Fnbottom-Fntop).
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what is the initial position, the bottom of the loop? assuming this, then we have
KE at bottom = PE + KE at top
KE at top = 1/2 m vbottom^2 - mg h
1/2 m vtop^2 = 1/2 m vbottom^2 - m g h
vtop^2 = vbottom^2 - 2 g h
vtop = Sqrt[(22.18m/s)^2 - 2 x 9.8m/s/s x 20m] = 10m/s
we use newton's second law to find the normal force at top and bottom
in each case, the forces are the normal force acting up and gravity acting down; they comvbine to produce a centrip force that points to the center of the loop
at the top:
N - mg = - m vtop^2/r
at the bottom
N - mg = + mvbottom^2/r
N bottom - N top = m vbottom^2/r + m vtop^2/r = m/r( v bottom^2 - v top^2)
subsittute m = 60kg, r = 10m, vtop = 10m/s and vbottom = 22.18m/s and solve for the difference in normal force
KE at bottom = PE + KE at top
KE at top = 1/2 m vbottom^2 - mg h
1/2 m vtop^2 = 1/2 m vbottom^2 - m g h
vtop^2 = vbottom^2 - 2 g h
vtop = Sqrt[(22.18m/s)^2 - 2 x 9.8m/s/s x 20m] = 10m/s
we use newton's second law to find the normal force at top and bottom
in each case, the forces are the normal force acting up and gravity acting down; they comvbine to produce a centrip force that points to the center of the loop
at the top:
N - mg = - m vtop^2/r
at the bottom
N - mg = + mvbottom^2/r
N bottom - N top = m vbottom^2/r + m vtop^2/r = m/r( v bottom^2 - v top^2)
subsittute m = 60kg, r = 10m, vtop = 10m/s and vbottom = 22.18m/s and solve for the difference in normal force
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'Check' it would imply that you post an answer and we see if it's right... ;-)
I'll give you a couple of hints.
To find the velocity at the top, remember that kinetic energy at the bottom is converted to gravitational potential energy at the top. Calculate the gravitational potential at the top in terms of 'm': we don't know the mass, but it ends up cancelling out anyway. Now calculate how much kinetic energy would correspond to that potential energy. Use that to calculate the velocity the cart would need just to get it to the top. Subtract *that* velocity from 22.18 m/s and you will have the velocity at the top of the loop.
For circular motion the centripetal acceleration is given by a = v^2/r, and to find the centripetal force remember F = ma. The weight force of a person is just F = mg. The normal force on the person will be the sum of the centripetal force and the weight force at the bottom of the circle and the weight force minus the centripetal force at the top of the circle.
I'll give you a couple of hints.
To find the velocity at the top, remember that kinetic energy at the bottom is converted to gravitational potential energy at the top. Calculate the gravitational potential at the top in terms of 'm': we don't know the mass, but it ends up cancelling out anyway. Now calculate how much kinetic energy would correspond to that potential energy. Use that to calculate the velocity the cart would need just to get it to the top. Subtract *that* velocity from 22.18 m/s and you will have the velocity at the top of the loop.
For circular motion the centripetal acceleration is given by a = v^2/r, and to find the centripetal force remember F = ma. The weight force of a person is just F = mg. The normal force on the person will be the sum of the centripetal force and the weight force at the bottom of the circle and the weight force minus the centripetal force at the top of the circle.