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Mechanics: Motion on Incline, Banking on Tracks, Projectile.

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Solution Summary

Three problems.
(1) the angle by which a man will lean forward while slipping on an frictionless incline is calculated.
(2) the banking of a circular race track and the force on a car on the track are calculated.
(3) the horizontal distance and the maximum height of a projectile is calculated.

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Here is the solution attached.

a) Determine the angle of the incline?

The angle of the incline  can be determined by solving the right angled triangle with perpendicular equal to 100 ft and the base equal to 200 ft as

Or

b) What angle would the man have to lean forward to, in order to lose his center of gravity and fall down the slope?

I think it should be "in order not to lose his center of gravity and fall down the slope?"

As there is no friction, the forces acting on the man are his weight mg and the normal force of the incline N. the weight has no torque about his center of gravity but the normal reaction may have. To remain standing on the incline while slipping and not to topple the torque due to the normal force N must also be zero and for this he must remain normal to the incline and hence incline with the vertical by the same angle  as the slopes makes with horizontal. Hence to remain without falling on the slope he must lean forward by angle  = 26.5650 with the vertical. If the angle is less then this he will fall behind and if the angle is more will fall forward.

c) Assuming the man did fall and rolled into a ball, what speed or velocity would the man be rolling at the bottom of the slope?

As there is no force other then the component of his weight along the incline, and the man rolled in to a sphere, his center of mass will accelerate with g*sin and will acquire a kinetic energy equal to the potential energy lost and hence the speed of center of mass is given by applying the law of ...

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