A steep mountain is inclined 74 degrees to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 870 ft from the base to the top of the mountain. Find the shortest length of cable needed. Include appropriate units in your answer.

An air bubble of volume 20 cm³ is at the bottom of a lake 40 m deep, where the temperature is 4.0°c. the bubble rises to the surface, which is at a temperature of 20°c.take the temperature of the bubble’s air to be the same as that of the surrounding water. just as the bubble reaches the surface, what is its volume?

Consider a double atwood machine constructed as follows: a mass 4m is suspended from a string that passes over a massless pulley on frictionless bearings. the other end of this string supports a second similar pulley, over which passes a second string supporting a mass of 3m at one end and m at the other. using two suitable generalized coordinates, set up the lagrangian and use the lagrange equations to find the acceleration of the mass 4m when the system is released. explain why the top pulley rotates even though it carries equal weights on each side.

14. a body is projected with velocity vi from a.at the same time another body is projectedvertically upwards from point b withvelocity v2 lies vertically below the highestpoint. both the bodies collide thenv2÷v1is