10.2 Fields at Work

This question is about forces and work done on objects held in fields. The first part is about charged oil drops in electric fields and the second part about space craft moving in gravitational fields.

Two charged, horizontal, parallel metal plates are a distance d apart.

A small oil drop P is positioned between the plates such that when the potential difference between the plates is V₁, the drop is stationary. The potential difference is changed to V₂ and the drop moves upwards with a constant velocity v.

Complete the diagrams with the names, directions and relative magnitudes of the forces acting on the oil drop for the situations when pd = V₁ and pd = V₂.

When a small, smooth sphere moves through a fluid such as air with low velocity v it experiences a resistive force. The force can be expressed in terms of a constant k so that:

The magnitude of the charge on the oil drop is q, and the distance between the plates is d.

Determine an expression which relates the velocity the oil drop moves upwards when the potential difference between the plates changes to the constant k. 

Work is done on the Soyuz spacecraft which is used to transport astronauts to the International Space Station (ISS). The ISS orbits the Earth at a height of 400 km above the Earth’s surface.  The Soyuz spacecraft has a mass of 7150 kg.

• Mass of the Earth = 5.97 × 10²⁴ kg
• Mean radius of the Earth = 6.37 × 10⁶ m

 For one trip from Earth to the ISS:

A binary planet system consists of two stars, A and B.

A has a mass of mass 4.0 × 10³⁰ kg and B has a mass of 8.0 × 10³⁰ kg. The centres of the stars are separated by a distance of 2 × 10⁸ km.

Calculate the gravitational potential at the midpoint between the stars.

The amount of energy required to send a space probe of mass 1800 kg from the surface of star A to the midpoint between stars A and B is 4.2 × 10¹¹ J.

Calculate the gravitational field strength on the surface of star A.

The two stars are drifting apart.

Calculate how far star A will have drifted at the point where its gravitational potential energy has decreased by 10 %.

The orbits of the Earth and Jupiter are very nearly circular, with radii of 150 × 10⁹ m and 778 × 10⁹ m respectively. It takes Jupiter 11.8 years to complete a full orbit of the Sun.

Show that the values in this question are consistent with Kepler’s third law.

Data from the orbits of different planets around our Sun is plotted in a graph of log(T²) against log(R³) as shown in the graph below, where T is the orbital period and R is the radius of the planet's orbit.

The values of T and R have been squared and cubed respectively due to Kepler's Third Law stating that:

Calculate the percentage error for the mass of the Sun obtained from the graph.

Evaluate this statement of Newton's law of gravitation: "The gravitational force between two masses is proportional to the masses and inversely proportional to the square of the distance between them."

A satellite of mass m orbits a planet of mass M. If the orbital radius is R and the orbital period is T, show that the ratio is constant.   

Calculate the change in gravitational potential energy of the satellite, of mass 39 kg, as it moves from an orbit of height 1100 km above the Earth's surface to one of height 2100 km.

• Mass of Earth = 6.0 × 10²⁴ kg
• Average radius of Earth = 6.4 × 10⁶ m

Explain whether the gravitational potential energy has increased, decreased or stayed the same when the orbit changes as in part (c). 

Define electric potential at a point in an electric field. 

A point charge of mass 1.30 × 10^–4 kg is moving radially towards a small, charged metal sphere as shown. 

The electric potential at the surface of the sphere is 9.00 × 10⁴ V. Determine if the point charge will collide with the metal sphere. 

Determine the speed at which the point charge is certain to collide with the metal sphere.

Protons are positively charged and are often described as "colliding" in particle accelerator experiments, as well as in the core of stars. 

Discuss the implications of two protons colliding in terms of the forces between them. Describe the conditions necessary for such a collision to take place. 

A charge –q with mass m orbits a stationary charge q with a constant orbital radius r. 

Draw the electrostatic force on –q due to the electric field created by q.

Show that the orbital speed of v is given by: 

Show that the total energy E of the orbiting charge is given by: 

Hence, determine an equation for how much energy must be supplied to –q if it is to orbit the stationary charge q at twice the radius in part (c), 2r.

Binary star systems involve two stars that orbit a common centre of gravity. One such system is shown. 

Each star has a mass M and orbital radius r, such that their separation is 2r. 

Deduce that the time period T of each star's orbit is related to the orbital radius r by the following equation:

Show that the kinetic energy of each star in the binary system is given by: 

Hence, show that the total energy of the binary star system is given by the equation: 

The binary system radiates energy in the form of gravitational waves.