Classical Mechanics

Conservation of Energy

Conservative Forces - Problem Solving

         

Four identical balls roll down from the tops of four different triangles to the ground, along the solid lines shown in the figures above. If the heights of all four triangles are equal, to which of the four balls does gravity deliver the most work?

A ball of mass m=5 kg  m = 5 \text{ kg } slides down from the top of a hemisphere of radius R=6 m . R = 6 \text{ m }. The speed of the ball when it loses contact with the hemisphere can be expressed as v=ab m/s, v = \sqrt{\frac{a}{b}} \text{ m/s}, where aa and bb are coprime positive integers. Find the value of a+b.a+b.

The gravitational acceleration is g=10 m/s2. g= 10 \text{ m/s}^2.

A ball of mass m=7 kg m = 7 \text{ kg} falls towards the ground from a height of H=8000 km. H = 8000 \text{ km} . When the ball's height is h=3000 km, h = 3000 \text{ km} , its speed can be expressed as v=GM×ab km/s, v = \sqrt{\frac{GM\times a}{b}} \text{ km/s}, where aa and bb are coprime positive integers. If the air resistance is negligible, what is the value of a+b?a+b?

GG is gravitational constant and MM is the mass of the earth. The radius of the earth is R=6000 km. R = 6000 \text{ km} .

A ball of mass m=5 kg  m = 5 \text{ kg } slides down from the top of a frictionless hemisphere of radius R=12 m. R = 12 \text{ m} . At what height hh does the ball lose contact with the hemisphere?

The gravitational acceleration is g=10 m/s2. g= 10 \text{ m/s}^2.

A ball of mass m=5 kg m = 5 \text{ kg} falls down towards the ground from a height of h=3000 km. h = 3000 \text{ km}. The speed with which the ball strikes the ground is v=GM×ab km/s, v = \sqrt{\frac{GM\times{a}}{b}} \text{ km/s}, where aa and bb are coprime positive integers. If the air resistance is negligible, what is the value of a+b?a+b?

GG is the gravitational constant and MM is the mass of the earth. The radius of the earth is R=6000 km. R = 6000 \text{ km} .

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