Applications of definite integral

9000072905

Level: 
C
The \(1000\, \mathrm{kg}\) heavy satellite is transported to the orbit \(150\, \mathrm{km}\) above the ground. Find the mechanical work required for this transport. The mass of the Earth is \(M = 6\cdot 10^{24}\, \mathrm{kg}\), gravitational constant \(\kappa = 6.67\cdot 10^{-11}\, \mathrm{N\, m}^{2}\mathrm{kg}^{-2}\) and Earth radius \(R = 6\: 370\, \mathrm{km}\). Round your result to nearest \(\mathrm{MJ}\).
\(1\: 445\, \mathrm{MJ}\)
\(1\: 471\, \mathrm{MJ}\)
\(1\: 412\, \mathrm{MJ}\)

9000072906

Level: 
C
The reservoir in the form of a box is filled with the water. The vertical side of the box is \(50\, \mathrm{cm}\) height and \(40\, \mathrm{cm}\) long. Find the total force which acts on this side. The mass density of the water is \(1\: 000\, \mathrm{kg\, m}^{-3}\) and the standard acceleration of gravity is \(g = 9.81\, \mathrm{m\, s}^{-2}\).
\(490.5\, \mathrm{N}\)
\(981\, \mathrm{N}\)
\(245.25\, \mathrm{N}\)

9000072907

Level: 
C
A homogeneous cube with the side \(10\, \mathrm{cm}\) is in the water. The bottom side is parallel to the water surface \(10\, \mathrm{cm}\) below the surface. Find the work required to move the cube to the position when the bottom side just touches the surface of the water. The mass density of the cube is \(2\: 000\, \mathrm{kg\, m}^{-3}\), the mass density of the water is \(1\: 000\, \mathrm{kg\, m}^{-3}\) and the standard acceleration of gravity is \(g = 10\, \mathrm{m\, s}^{-2}\).
\(1.5\, \mathrm{J}\)
\(2\, \mathrm{J}\)
\(1\, \mathrm{J}\)

9000072908

Level: 
C
A \(100\, \mathrm{kg}\) heavy anchor is attached to a \(20\, \mathrm{m}\) long rope. One meter of the rope weights \(1\, \mathrm{kg}\). Find the work required to raise the anchor \(20\, \mathrm{m}\) higher. The standard acceleration of gravity is \(9.81\, \mathrm{m\, s}^{-2}\). Neglect the buoyancy (the force from the Archimedes law).
\(21\: 582\, \mathrm{J}\)
\(23\: 544\, \mathrm{J}\)
\(19\: 620\, \mathrm{J}\)