# Newton's laws --- Introduction ---

This module currently includes 19 simple exercices on Newton's laws :
• Motion on an inclined plane : Frictionless sliding, Constant velocity sliding
• Problems with two masses : Blocks in contact, Masses connected by a taut wire, Stacked blocks
• Mass attached to a spring : Animated important QCM
• Object in a moving frame of reference : Object suspended in an elevator, Package on a train, Pendulum on a train, Mass placed on a turntable

• Many exercises involve the force of friction between two solid objects - They are identifiable by a star following its title: *
• Numerical answers must be given with a precise number of significant digits. To get your students familiar with such a notion, an exercice ("Significant digits") is proposed.
• In some exercises, suggestions or course reminders are given via the link "help" at the bottom of the statement

### Tension of a simple pendulum string

 A pendulum, consisting of a mass m= kg that is suspended from a string l= m long, is set free from an initial position having an angle ° with respect to the vertical. ( ) What is the modulus, V, of the mass velocity when the pendulum passes through its vertical position? V (m/s)= Deduce the string tension, T, at the same instant: T(N)= Assume g=9.81 m.s-2 and give the answers with 2 significant digits. xrange -1.2,1.2 yrange -1.3,0.4 frect -0.1,0,0.1,0.05,black line 0,0,-sin(*pi/180),-cos(*pi/180),black fcircle -sin(*pi/180),-cos(*pi/180),16,red arc 0,0,.3,.3,270-,270,black text black,,-.3,medium, dline 0,0,0,-0.9,black circle 0,-1,16,red arrow 0,-1,0.3,-1,10,blue text blue,0.1,-1.1,medium,V ? text black,0.1,-0.5,medium,T ?

### Will the package slip?

A box filled with books, having a mass kg, rests on the ground.
Let = and = respectively be the static and dynamic friction coefficients between the cardboard and the ground.
An horizontal force of F= Newton is applied on the box.
Determine the modulus of the friction force acting on the box (rounded to the nearest Newton).
(Assume g= 10 m/s2)

xrange 0,150 yrange 0,100 dline 5,5,150,5,black rect 50,6,145,80, blue arrow 5,25,50,25,12,red text red, 10,20,large, F

It is assumed that in the conditions of the experiment, the equilibrium of the box can only be broken by slip (the box will not overturn).

( )

### Sliding with friction on the horizontal plane

A block B of mass mB= g is suspended from a wire of negligible mass
which passes over an (ideal) pulley and is then connected to a block A of mass mA,
resting on a horizontal table.

Let = and = respectively be the static and dynamic friction coefficients of block A on the table surface and let us assume g=9.81 m/s2.

xrange 0,2 yrange 0,1 line 0.05,0.7,1.6,0.7,black rect 0.05,0.71,0.25,0.91, blue line 0.25,0.8,1.7,0.8, black circle 1.7,0.7,22, black line 1.8,0.7,1.8,0.4, black rect 1.7,0.4,1.9,0.2, red text blue, 0.1,0.9, large, A text red, 1.8,0.35, large, B

The ensemble will slide if mA is at : g .
(The answer must be rounded to the gram).

Knowing that mA= g, determine with 2 significant digits,
• the modulus of the acceleration, a, of the ensemble : a = m/s2
• the tension, T, of the wire : T = N

### Identical masses connected by a taut wire - QCM 1

xrange 0,2 yrange 0,1 line 0.05,0.7,1.6,0.7,black rect 0.05,0.71,0.25,0.91, blue line 0.25,0.8,1.7,0.8, black circle 1.7,0.7,22, black line 1.8,0.7,1.8,0.4, black rect 1.7,0.4,1.9,0.2, red text blue, 0.1,0.9, large, A text red, 1.8,0.35, large, B

A block B of mass m is suspended from a wire of negligible mass which passes over an (ideal) pulley and is then connected to a block A having the same mass m, resting on a long horizontal table.
We neglect friction forces between the block A and the table.
As long as the block A does not get in contact with the pulley, we can deduce that :

### Sliding at a constant velocity - QCM

 A package slides along an inclined plane with a constant velocity. It is known that the total force exerted by the inclined plane is the sum of the reaction force, normal to the incline, and the friction force parallel to the incline It is possible to deduce : xrange -10,100 yrange -10,100 line 0,0,60,0,black line 0,0,60,30, black line 30,15,52,26,red line 52,26,41,49,red line 41,49,19,37,red line 19,37,30,15,red arrow 35.7,31.8,8.8,18.4,12,blue

### Frictionless sliding - QCM

 You are skiing down a piste when all of the sudden one of your skiis, having a mass M, dettaches and slids downhill with almost no friction. It is possible to deduce that: xrange 0,1 yrange 0,1 line 0,0,0.99,0.4, black line 0.6,0.25,0.9,0.37, red line 0.6,0.26,0.9,0.38, red arc 0.6,0.3,0.1,0.1,225,270, red arc 0.6,0.31,0.1,0.1,225,270, red text red,0.6,0.5,small,ski arc 0,0,0.5,0.5,0,22, black ellipse 0.35,0.05,0.06,0.10, black line 0.32,0.05,0.38,0.05, black

### Blocks in contact - QCM1

Two carboard boxes full of books are in contact with each other and both of them rest on the ground.
The friction forces between the carboard and the ground are negligible.
The mass of the box B is equal to times the mass of the box A.
You push the box A by applying a horizontal force F to it.

xrange 0,6 yrange 0,1 dline 1,0.05,6,0.05,black rect 2.5,0.06,3.5,0.6, blue rect 3.51,0.06,5,0.95, red arrow 1.2,0.3,2.45,0.3,12,black text black, 1.6,0.7,large, F text blue, 3,0.5, large, A text red, 4.2,0.8, large, B

The resultant force on the box B is equal to... (tick your answer below)

### Hanging object inside an elevator - QCM2

A decorative object of mass M is hanging from a string connected to the ceiling of an elevator cab.
• The elevator, initially at rest, is going to move down more and more quickly.
At the initial instant, the tension of the string .
When the elevator descent reaches its cruising velocity the elevator acceleration becomes null.
From this moment on, the string tension .

• Let us now imagine a new situation when the elevator, initially at rest again, is going to to ascend with increasing speed.
At the initial instant, the string tension .
Once the cruising velocity is attained the elevator acceleration becomes null.
From this moment on, the string tension .

• The elevator was rising at constant velocity when all of the sudden the cable that supported it broke.
During the elevator free fall the tension of the string whereof the object is suspended:

### Blocks in contact - QCM2

Two carboard boxes full of books are in contact with each other and both of them rest on the ground. Let be the dynamic friction coefficient between the cardboard and the ground. The mass of the box B is equal to times the mass, m, of the box A. You push the box A by applying to it a horizontal force F such that the boxes move with a constant velocity.

xrange 0,6 yrange 0,1 dline 1,0.05,6,0.05,black rect 2.5,0.06,3.5,0.6, blue rect 3.51,0.06,5,0.95, red arrow 1.2,0.3,2.45,0.3,12,black text black, 1.6,0.7,large, F text blue, 3,0.5, large, A text red, 4.2,0.8, large, B

### Stacked blocks - QCM1

Two cardboard boxes full of books are stacked one on the other.
Let be the static friction coefficient between the two cardboard surfaces and let us neglect friction between the bottom carboard and the ground.
The mass of the box A is times the mass, m, of the box B. You push the box A by applying a horizontal force F to it.

xrange 0,200 yrange 0,100 dline 5,5,200,5,black rect 70,6,170,35, blue rect 80,36,160,98, red arrow 10,25,70,25,12,black text black, 23,22,large, F text blue, 115,25, large, A text red, 115,80, large, B

N.B.: We assume the weak friction conditions where B will not tip over A.

The two boxes will not slide with respect to one another if...?

### Stacked blocks - QCM2

 Two cardboard boxes full of books are stacked one on the other. Let s be the static friction coefficient between the two cardboard surfaces and let us neglect friction between the bottom carboard and the ground. The mass of the box A is times the mass, m, of the box B. You push the box B by applying a horizontal force F to it. N.B.: We assume the weak friction conditions where B will not tip over A. The two boxes will not slide with respect to one another if (Tick your answer below ): xrange 0,200 yrange 0,100 dline 5,5,200,5,black rect 50,6,150,35, blue rect 60,36,140,98, red arrow 10,50,60,50,12,black text black, 20,40,large, F text blue, 95,25, large, A text red, 95,80, large, B

### Masses connected by a taut wire - QCM3

 A block A of mass m is at rest on a horizontal table. It is connected by a wire (of negligible mass) which passes over an (ideal) pulley to a block B of mass m/. Lete be the static friction coefficient between the block A and the table. We assume that under the conditions of the experiment the equilibrium of the system can only be broken by sliding (the block A cannot tip over) The system will be in equilibrium if and only if (tick you answer below): xrange 0,150 yrange 0,100 line 5,70,100,70,black rect 10,71,70,99, blue line 70,80,110,80, black circle 110,70,20, black line 120,70,120,40, black rect 105,40,135,20, red text blue, 42,90, large, A text red, 118,33, large, B

### Masses connected by a taut wire - QCM2

 A block A of mass m is at rest on a horizontal table. It is connected by a wire (of negligible mass) which passes over an (ideal) pulley to a block B of mass . Let us neglect friction forces between the block A and the table. The modulus of the acceleration of the block B is given by: The tension of the wire is given by: xrange 0,2 yrange 0,1 line 0.05,0.7,1.6,0.7,black rect 0.05,0.71,0.25,0.99, blue line 0.25,0.8,1.7,0.8, black circle 1.7,0.7,22, black line 1.8,0.7,1.8,0.4, black rect 1.7,0.4,1.9,0.2, red text blue, 0.1,0.9, large, A text red, 1.8,0.35, large, B

### Package in a train

A suitcase of mass 11 kg sits on the floor in the middle of the hallway of a car, of lengh 2L= m, which is moving horizontally at a constant velocity along the direction of the 0x axis.
xrange 0,200 yrange 0,100 arrow 5,10,190,10,10,black rect 10,12,130,80, blue frect 65,13,75,23, red arrow 70,35,15,35,10,black arrow 70,35,125,35,10,black text black, 60,50,small, 2L text black, 150,25,small, axe Ox text black, 5,90,small, backwards text black, 120,90,small, forwards
Let s= and d= respectively be the coefficients of static and dynamic friction between the suitcase and the floor, and let us assume g= 10 m/s2. At the initial instant t=0, the speed of the car with a constant of modulus A.

The block will not slide on the floor if A does not exceed the value A0 (m/s2)=

Let us assume that we actually have A=*A0.
The block will slip with respect to the floor and will hit one of the car vertical walls
after a time t (s)= (give 2 significant digits).

### Pendulum in a train

A pendulum of mass g is suspended from the ceiling in the middle of a car that is moving horizontally at a constant velocity along the direction of the 0x axis.
The height of the car is equal to m and the vertical of the pendulum intersects the car floor at point H.
Let us assume g=10 m/s2

xrange 0,200 yrange 0,100 arrow 5,10,190,10,10,black rect 10,12,130,80, blue line 70,80,70,50,black fcircle 70,45,10,red dline 70,40,70,10,black text black, 150,25,small, axe Ox text black, 5,90,small, arriere text black, 120,90,small, avant text blue, 75,22,small, H fcircle 70,12,4,blue

At the initial instant, , the car speed begins to with a constant of modulus A = g.

The pendulum will therefore tilt with respect to the train,
defining an angle with respect to the vertical (give the answer rounded to the degree).

If at a given time we cut the pendulum string
the pendulum will hit the car floor at a distance d from point H, d = (give the answer in m and use 2 significant digits). ( )

### Mass placed on a turntable

A block of mass m rests on an horizontal disk that is rotating anticlockwise with a constant angular velocity .
The block does not slide on the disk and it is placed at a distance L from the center of rotation O.
Let be the friction force exerted by the disk on the block, the static friction coefficient between the block and the disk, and we are going to use .

xrange 0,180 yrange 0,180 circle 80,90,120,blue 80,90,137,147,10, 80,90, 24,147,10, 80,90,160, 90,10, 80,90, 80,170,10, text , 20,160, small, R' text , 90,170, small, R text black, 75, 85, small, O arc 80,90,140,140,280,315,blue line 130,40,120,40,blue line 128,40,128,33,blue text blue,110,30,small, rotation text blue,100,20,small, of the disk fcircle 108,118,10, red text red, 110,110,smal,m text black, 95,105,small,L
It is possible to deduce (tick ALL of the correct answers) :

### Significant digits

Rounding a numerical result to a given number of significant digits.
( )
• Write the number: using :
then using significant digits:

• Write the number: using :
then using significant digits:

• Write the number: using :
then using significant digits:

• Write the number: using :
then using significant digits:

### Mass attached to a string - QCM

animate 24,1,0 xrange -0.75,0.45 yrange line -0.1,0,0.1,,red line 0.1,,-0.1,,red line -0.1,,0.1,,red line 0.1,,-0.1,,red line -0.1,,0.1,,red line 0.1,,-0.1,,red line -0.1,,+0.1,,red line 0.1,,0,,red dline -0.42,,0,,black arrow ,,,,8, arrow ,,,,8, arrow ,,,0.6,8, arrow 0.4,0,0.4,,8,black arrow 0.4,,0.4,0,8,black text black,0.3,,medium,L arrow -0.6,0,-0.6,,8,black arrow -0.6,,-0.6,0,8,black text black,-0.7,,medium,b fcircle 0,,14,blue frect -0.4,-0.03,0.4,0,black arrow ,10,black text black, -0.5,,medium,0 text black, -0.5,0.6,medium,z text black, -0.47,,small,e line -0.42,,-0.38,,black line -0.42,0.6,-0.38,0.6,black text black, -0.38,,medium,z

A block of mass m is attached to a spring having a constant equal to k and a natural length (at rest) equal to b.
This block oscillates vertically around its equilibrium position, ze.

• Its weight is represented by the vector :
• The elastic force due to the spring is represented by the vector :
• and the resultant force acting on the block is represented by the vector :

For a given position of the mass, represented by its abscissa, z, write an expression for the following quantities as a function of m, g, k, b and z :

• weight :
• elastic force :
• The equilibrium position, ze, of the mass

### Passenger at subway startup

A man falls backwards when the subway starts up.

If we work (we place ourselves) in the frame of reference , which one of the following three is a correct description of the horizontal forces acting on the passenger ?

Explain what is the nature of the represented forces :

• La force est
• La force est
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