Suppose a straight tube could be drilled right through the Earth, modelled as a uniform sphere of radius R and total mass M. A ball of unit mass is dropped into the tube at the surface of the Earth.
Suppose a straight tube could be drilled right through the Earth, modelled as a uniform sphere of radius R and total mass M. A ball of unit mass is dropped into the tube at the surface of the Earth.
A body of unit mass inside a uniform sphere at a distance r from its centre O experiences a gravitational attraction towards the centre ofwhere G is the gravitational constant and Mr is the mass of material inside the sphere of radius r.
(In other words it is as if the body were on the surface of a sphere of radius r, all the matter further from the centre than r will have no net gravitational effect on the body.)
Suppose a straight tube could be drilled right through the Earth, modelled as a uniform sphere of radius R and total mass M. A ball of unit mass is dropped into the tube at the surface of the Earth.
i) Work out the mass of a sphere of the Earth of radius r, in terms of R, M and r.
ii) Homework help – Write down an expression for the gravitational force on the ball when at a distance r from the centre.
iii) Hence show that the ball will oscillate with simple harmonic motion with an amplitude equal to the radius of the Earth. Determine the period.
Assume that a straight tube, described as a homogeneous sphere of radius R and total mass M, could be dug directly through the Earth. A unit mass ball is put into the tube near the Earth’s surface.
Assume that a straight tube, described as a homogeneous sphere of radius R and total mass M, could be dug directly through the Earth. A unit mass ball is put into the tube near the Earth’s surface.
A body with unit mass located at r from the center of a uniform sphere. Where G is the gravitational constant and Mr is the mass of material inside the sphere of radius r, O experiences a gravitational attraction towards the center of.
(To put it another way, it
