Rolling sphere on incline

As shown in the figure a uniform solid sphere rolls on a horizontal surface at 30 m/s and then rolls up the incline. If frictional losses are negligible, what will be the value of its height when the it stops ?

This is a classical problem in mechanics. An object is let to slide on a frictionless sphere. In what angle the object will break free form the sphere?

An object is positioned at an initial angle on a frictionless spherical shell. It is then let go, and starts sliding on the shell. In what final angle the object will break free from the shell? the solution should be in terms of the initial angle.

A compressed spring with a stone on it, flings the stone upward when released. Find the maximum height attained.

A spring is .66 m long when it contains no energy When a stone with mass M= 8.5 kg is placed on the spring it shortens by b= .06 m then supports the stone in equilibrium. The stone is now pushed down an additional d= .38 m. When released, the stone is projected upward to maximum height H meters above release point. SEE ATTACHMENT #1 for a diagram of parameters of the event. Find H, the maximum upward movement of the stone.

Two springs can be combined in series or in parallel. Find the force constant of each combination.

Spring A has force constant kA= 60 nt/m. Spring B has force constant kB= 40 nt/m. The two springs attached end to end form a series combination with force constant Ks. The two springs attached side by side so that a force applied to the ends stretches both springs, form a parallel combination with force constant Kp. SEE ATTACHMENT #1 for a diagram of the springs and both arrangements. PART a. Find Ks, the force constant of the series combination. PART b. Find Kp, the force constant of the parallel combination.

A non linear force does work. By integration, find the work done between given limits.

A perfectly elastic rubber framework is stretched by a force applied to its end. This force as a function of the x coordinate of the end is: (1) F = 52.8 x + 38.4 x^2 PART a. By integration, find the work done by this applied force from position x1=.5 m to x2= 1 m. PART b. With the end at position x2, a frictionless mass of M= 2.17 kg is attached to the end. Upon release the mass is moved to x3 = .75 m. Find the speed of the mass at that point.