Angular momentum and central forces

In a hospital laboratory, a centrifuge spins a test tube of $0,02\ \text{kg}$ that describes a circle of radius $0,15\ \text{m}$ with a speed of $9\ \text{m/s}$. The diagram shows the rotation viewed from above:

1. a) Write down the expression for the angular momentum of a particle about a point, $L = m \cdot v \cdot r$, valid when the velocity is perpendicular to the radius.
2. b) Calculate the angular momentum of the tube about the axis of rotation, expressed in $\text{kg} \cdot \text{m}^2/\text{s}$.

An athlete whirls a sling: a stone of $0,1\ \text{kg}$ describes a horizontal circle of radius $0,8\ \text{m}$ with a speed of $12\ \text{m/s}$.

1. a) Calculate the linear momentum of the stone, $p = m \cdot v$.
2. b) Calculate its angular momentum about the centre of the circle, $L = m \cdot v \cdot r$.

A nanosatellite of $4\ \text{kg}$ describes a circular orbit of radius $7 \cdot 10^{6}\ \text{m}$ around the Earth with a speed of $7,5 \cdot 10^{3}\ \text{m/s}$. The diagram shows the orbit:

1. a) Calculate its angular momentum about the centre of the Earth, expressing the result in standard form.
2. b) State the direction and sense of the vector $\vv{L}$ relative to the plane of the orbit.

The Orion spacecraft, with a mass of $2,5 \cdot 10^{4}\ \text{kg}$, passes through a point $2 \cdot 10^{6}\ \text{m}$ from the centre of the Moon with a speed of $1,5 \cdot 10^{3}\ \text{m/s}$, in a direction making an angle of $30^\circ$ with the line joining it to the Moon ($\sen 30^\circ = 0,5$). The figure shows that instant:

1. a) Write down the general expression for the magnitude of angular momentum, $L = m \cdot v \cdot r \cdot \sen\theta$, and explain the meaning of each term.
2. b) Calculate the angular momentum of the spacecraft about the centre of the Moon.
3. c) If the spacecraft were to pass through the same point with its velocity pointing directly towards the Moon, what would its angular momentum be? Reason your answer.

In the hammer throw event, the ball of $7,26\ \text{kg}$ rotates with a speed of $25\ \text{m/s}$ and its angular momentum about the axis of rotation is $290\ \text{kg} \cdot \text{m}^2/\text{s}$.

1. a) Rearrange the expression $L = m \cdot v \cdot r$ to solve for the radius of rotation.
2. b) Calculate the radius of rotation of the ball, to two significant figures.
3. c) Compare the result with the regulation length of the wire, which is about $1,2\ \text{m}$, and explain the reason for the difference.

Two test bodies, A and B, rotate in the laboratory about the same axis. Body A has a mass of $0,3\ \text{kg}$, a speed of $4\ \text{m/s}$, and a radius of rotation of $0,5\ \text{m}$; body B has a mass of $0,6\ \text{kg}$, a speed of $2\ \text{m/s}$, and a radius of $1\ \text{m}$. The diagram shows both bodies viewed from above:

1. a) Calculate the angular momentum of each body about the axis.
2. b) Calculate the ratio $\frac{L_B}{L_A}$ and interpret it: how many times greater is the angular momentum of B?
3. c) Without repeating calculations, reason how this ratio would change if the speed of A were doubled.

A force is central when its line of action always passes through a fixed point called the centre of force. State which of the following forces are central and which are not, justifying each answer:

1. a) The gravitational attraction of the Sun on a planet.
2. b) The tension in the string of a ball spinning attached to a pole.
3. c) The electric force exerted by the nucleus of an atom on an electron.
4. d) Air resistance acting on an artificial satellite in low orbit.
5. e) The thrust of a rocket engine accelerating in the direction of motion.

In the Bohr model of the atom, the electron of a hydrogen atom describes a circular orbit of radius $5,3 \cdot 10^{-11}\ \text{m}$ with a speed of $2,2 \cdot 10^{6}\ \text{m/s}$ (mass of the electron: $9,1 \cdot 10^{-31}\ \text{kg}$). The diagram represents the model:

1. a) Calculate the angular momentum of the electron about the nucleus, in standard form and to two significant figures.
2. b) State which force acts as the centripetal force in this case and justify that it is a central force.

Two forces of $20\ \text{N}$ are applied to a rod that can rotate about an axis, at points located $0,4\ \text{m}$ from the axis: force A points radially towards the axis and force B is perpendicular to the rod ($\sen 90^\circ = 1$, $\sen 0^\circ = 0$). The diagram shows the rod and the two forces:

1. a) Calculate the moment of each force about the axis, $M = r \cdot F \cdot \sen\theta$.
2. b) From $\frac{\Delta \vv{L}}{\Delta t} = \vv{M}$, state which of the two forces changes the angular momentum of the rod and which does not.
3. c) Relate the result to the behaviour of central forces: which of the two forces does a central force resemble?

It is desired to place an Earth observation satellite of $500\ \text{kg}$ into a circular orbit of radius $7 \cdot 10^{6}\ \text{m}$ measured from the centre of the Earth. Data: $G = 6,67 \cdot 10^{-11}\ \text{N} \cdot \text{m}^2/\text{kg}^2$ and $M_{Tierra} = 5,98 \cdot 10^{24}\ \text{kg}$.

1. a) The only force acting on the satellite is the gravitational force, which acts as a centripetal force. Equate $G \cdot \frac{M \cdot m}{r^2} = m \cdot \frac{v^2}{r}$ and deduce the expression $v = \sqrt{\frac{G \cdot M}{r}}$.
2. b) Calculate the orbital speed of the satellite.
3. c) Calculate its angular momentum about the centre of the Earth.
4. d) Justify, based on the central nature of the gravitational force, that this angular momentum remains constant throughout the orbit.

A proton ($m = 1,67 \cdot 10^{-27}\ \text{kg}$) moves in a straight line with constant speed $2 \cdot 10^{6}\ \text{m/s}$ inside the chamber of a particle accelerator. Its trajectory passes at a minimum distance $d = 3 \cdot 10^{-9}\ \text{m}$ from a sensor located at point O.

1. a) The angular momentum about O is $L = m \cdot v \cdot r \cdot \sen\theta$. Draw the situation in your notebook and reason that, although $r$ and $\theta$ change along the path, the product $r \cdot \sen\theta$ is always equal to $d$.
2. b) Calculate the angular momentum of the proton about O, in standard form.
3. c) The proton moves freely, without forces: is it consistent for its angular momentum to be constant even though it does not describe a curve? Justify your answer.

You can do it on paper and send a photo with your answer.

A probe of $800\ \text{kg}$ with its engines turned off approaches Mars. At the point of closest approach, it is at $4 \cdot 10^{6}\ \text{m}$ from the centre of the planet and travels with a speed of $5 \cdot 10^{3}\ \text{m/s}$, perpendicular to the line joining it to Mars.

1. a) Calculate the angular momentum of the probe at that point.
2. b) While the engines are turned off, the only force acting on the probe is the gravitational attraction of Mars: justify why the angular momentum remains constant throughout the manoeuvre.
3. c) When the probe moves away again and is at $1,6 \cdot 10^{7}\ \text{m}$ from the centre of Mars, moving perpendicular to the line joining them, calculate its speed.