Chapter 6

Mechanical point

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 ?

Hint

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 ?

Hint

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 ?

Hint

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 ?

Hint

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.

Hint

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 ?

Hint

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 ?

Hint

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  ?

Hint

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 ?

Hint

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  ?

Hint

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 ?

Hint

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) ?

Hint

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 ?

Hint

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  ?

Hint

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   ?

Hint

Solution

  • Moment of a force :

  • Angular momentum :

Question

Give Coulomb's law of dry friction.

Hint

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 ?

Hint

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 .

Hint

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.

Hint

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 .

Hint

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 ?

Hint

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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