Chapter 6

# To test the understanding of the lesson

## Question

• What is the principle of inertia ? (or first law of motion)

• What is Newton's third law ?

### Solution

• The center of mass of an isolated system is at rest or in a uniform motion (according to Galileo).

This principle holds true in an inertial frame of reference.

• According to Newton's third law, when one point exerts a force on a second point, the second point simultaneously exerts a force equal in magnitude and opposite in direction on the first point.

## Question

• What is an inertial frame of reference ?

• What is the difference between Kepler's frame of reference and Copernicus's frame of reference ?

• What is the geocentric frame of reference ?

### Solution

• An inertial frame of reference is one in which the principle of inertia holds true.

• Kepler's frame of reference : (heliocentric frame of reference)

Its origin is the center of mass of the Sun.

Its three axises are orientated towards three « fixed » stars.

• Copernicus's frame of reference :

Centered on the center of mass of the solar system.

• Geocentric frame of reference :

Its origin is the center of mass of the Earth.

Its three axises are orientated towards three « fixed » stars.

The geocentric frame is in circular translation relatively to Kepler's frame.

## Question

What are the possible motions of an isolated punctual object ?

At rest, rectilinear uniform translation, accelerated or deaccelerated ?

### Solution

According to the principle of inertia : at rest or in rectilinear uniform translation.

## Question

Does a force depend on which frame we calculate it in ?

### Solution

No : for instance, weight has always the same value, it does not depend on the chosen frame.

## Question

What is equivalent to the momentum in a circular rotation ?

Same question for the torque, angular momentum, force.

### Solution

It is the angular momentum :

which gives the moment of the momentum with respect to O.

It is the same equation between a force and its torque with respect to O :

## Question

Someone drops a package (mass ) from a plane flying horizontally with constant velocity.

How does this package drop with respect to the plane, if air friction is neglected ?

### Solution

The package keeps the same uniform rectilinear translation motion. It stays vertically to the plane.

## Question

Don't you find incredibly simple that the second law of motion only contains the second temporal derivative of the position, multiplied by a constant ?

Why doesn't it contain position and its derivatives until infinity, each one multiplied by a constant of appropriate dimension, like in the following equation ?

### Solution

•  : the existence of a force would then depend on the chosen frame of reference !

•  : a material point can have constant velocity without having a force exerted upon (first law of motion).

•  : if , acceleration is constant and the motion is accelerated : by experience, this is false.

## Question

What is the gravitational potential energy  ?

### Solution

Let be the horizontal axis of a punctual mass . If (Oz) is orientated upwards, , if it is orientated downwards, .

## Question

• What is the potential energy of a particle of mass in the gravitational field of the Earth (masse ) ?

• What is the electrostatic potentiel energy of two punctual charged particles and , the distance between the two being ?

### Solution

• This potential energy is , is the distance between the particle and the center of the Earth.

• The electrostatic potentiel energy of two punctual charged particles is :

## Question

What is the elastic potential energy of a spring, with spring constant  ?

### Solution

This energy is , where is the length of the spring and is the length when the spring is at rest.

## Question

What is a conservative force ? What about the work this force produces ? Why is mechanical energy conserved ?

### Solution

A conservative force derived from a potential energy :

The elementary work is :

It follows that the work of a conservative force is opposite to the variation of potential energy.

From the work energy theorem : (in an intertial frame of reference)

If is the mechanical energy of the particle :

It follow that mechanical energy is constant.

## Question

What is the centrifugal potential energy of a material point of mass in a non-inertial reference frame, rotating with constant angular velocity relatively to an axis (Oz) ?

### Solution

This energy is :

If is the distance between (Oz) and the point.

## Question

• What is the inertial force in a non-inertial frame of reference rotating with constant angular velocity relatively to an axis Oz ?

• What is the Coriolis force ?

### Solution

• Inertial force : , if H is the orthogonal projection of M on (Oz).

• Coriolis force : , if is the relative velocity of M.

## Question

What is the Lorentz force acting on a particle of charge , with velocity in an electromagnetic field  ?

### Solution

Lorentz force is :

## Question

• What is the moment with respect to O of a force applied on A ?

• What is the angular momentum with respect to O of a material point M of mass , moving with a velocity   ?

### Solution

• Moment of a force :

• Angular momentum :

## Question

Give Coulomb's law of dry friction.

### Solution

• Static friction :

and

• Kinetic friction :

and

## Question

What is the difference between the weight of a body and the gravitational pull that the Earth exerts on this body ?

### Solution

• The gravitational pull is :

• Weight is the resulting force of the gravitational pull and of the centrifugal force (the Earth spins with angular velocity around the North-South axis) :

If H is the projection of M on the North-South axis.

## Question

Give Kepler's third law, which contains the period and the radius of a uniform circular motion of a satellite around the Earth of mass .

### Solution

Kepler's third law :

## Question

• Give the general work energy theorem for power in a closed system.

• Give the work energy theorem for power in a closed solid system.

### Solution

• In an inertial reference frame, the work energy theorem for power in a closed system is :

• For a closed solid system, the power of exterior forces is equal to zero :

## Question

What is the minimum distance from which a driver must start braking at a red light if the speed of his car is  ?

The dry friction coefficient between the rires and the road is .

### Solution

The kinetic energy of the vehicle is converted in the work of friction force :

So :

## Question

Is the barycentric frame of reference always inertial ?

### Solution

It is inertial if the velocity vector of the center of mass is constant, it means if the system is isolated.

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