Physics exercises for the 1DF course

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Physics exercises for the 1DF course

Alice Gasparini

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1 Forces and interactions

• Exercise 1. Draw the force acting on the system Y represented by the point below.

The characteristics of this force are:

- horizontal direction;

- toward the right;

- intensity : F = 12N.

Scale: 1cm←→3N.

• Exercise 2. Draw the force acting on the system Z represented by the point below.

The characteristics of this force are:

- horizontal direction;

- toward the left;

- magnitude : F = 18N.

Scale: 1cm←→6N.

• Exercise 3. Give the characteristics of the force F~ acting on the system W represented in the figure below, knowing that the scale is1cm ←→6N.

• direction: ...

• orientation: ...

• intensity: ...

Exercises 3 Series 1

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• Exercise 4. Alain pulls a spring using the force F~ represented in the figure below.

What are the characteristics of this force?

Scale: 1cm ←→2N.

• Exercise 5. Anaïs pulls the dynamometer (“tendeur” T) with her hand (“main” M).

Draw (not in the figure) the forceF~ of the hand on the dynamometer, knowing that its intensity isF = 5N.Use the scale 1cm ←→2N.

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• Exercise 6. (In your exercise book) For each of the following situations, draw

- a sketch,

- a system-interaction diagram,

- a diagram of the forces (without scale) acting on the underlined object . 1. a cube hung by a spring, which is attached to the ceiling;

2. a flower-pot put on a table, standing on the floor;

3. a motorbike standing on the road (not moving);

4. a motorbike standing on the road, pushed by a man, but still not moving;

5. the Earth gravitating around the Sun and having the Moon as satellite (you can disregard the interactions due to the other planets or stars);

6. a space probe moving into space, far from any celestial body.

• Exercise 7. For each quantity, indicate whether it is a scalar (just a number with its unit) or a vector (number + direction + orientation), and write the relative SI unit.

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• Exercise 8. (In your exercise book)

a) A spring is stretched by two hands. The hands are moving symmetrically.

1. Draw the system - interaction diagram for the spring.

2. Draw the diagram of the forces acting on the spring (without scale).

b) The same spring is stretched by the same hands. The right hand moves while the left one is at rest.

1. Draw the system - interaction diagram for the spring.

2. Draw the diagram of the forces acting on the spring (without scale).

a) A marble made of iron is hung by a wire and a magnet is placed under the marble.

1. Draw the system - interaction diagram for the marble.

2. Draw the diagram of the forces acting on the marble (without scale).

b) What happens if the magnet is placed above the marble?

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2 Gravitation and weight

1. Calculate the magnitude of the attractive forces acting between two masses of 1,00kg at a distance d = 1,00m from each other. Then draw the situation and both the forces with the correct point of application (don’t forget to specify the scale used).

2. Starting from the result in 1, find the magnitude of the forces acting between the same two masses at a distanced⁰ = ₁₀^d from each other.

3. Starting from the previous results, find the magnitude of the forces acting between the same two masses at a distanced⁰⁰ = ₂₀^d from each other.

• Exercise 2. (In your exercise book, answer with 2 significant digits)

1. Calculate the magnitude of the gravitational force exerted by the Earth on the Moon. Is the magnitude of the gravitational force of the Moon on the Earth the same?

2. Calculate the magnitude of the gravitational force exerted by the Sun on the Earth;

3. Compare those two magnitudes calculating their ratio.

[ 20·10¹⁹N; 35·10²¹N; 1,7·10² ]

Calculate the magnitude of the gravitational forces acting between two tankers having masses m₁ = 1,5·10⁵tons and m₂ = 2,5·10⁵tons. The distance separating the centers of mass of the two tankers is120m.

[ 170N]

A satellite is at a heighth= 3,59·10⁷mabove the Earth’s surface. Its mass is1,12tons.

a) Calculate the distance between the center of gravity of the satellite and the center of the Earth.

b) Calculate the magnitude of the gravitational forces acting between the satellite and the Earth.

[42,3·10⁶m; 249N ]

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Calculate the distance between the center of gravity of a spaceship of 38,2tons and the surface of Neptune, knowing that the gravitational force of this planet on the spaceship has a magnitude of 345N (draw a sketch of the situation).

[ 847·10⁶m ]

Mu Arae d is an exoplanet revolving around the star Mu Arae. The radius of the orbit is estimated to be 1,5·10¹⁰mand the mass of the star 2,2·10³⁰kg. The magnitude of the gravitational forces between the two bodies is5,4·10²⁴N.

What is the mass of Mu Arae d? Compare it with the mass of the Earth by calculating the ratio between the two masses.

[ 8,3·10²⁴kg; 1,4 ]

• Exercise 7. (In your exercise book) Calculate:

1. the magnitude of the weight acting on a piece of chocolate of 93g;

2. the magnitude of the weight exerted on a cube of lead whose side measure 50mm;

3. the magnitude of the weight on an astronaut of 130kg on the Moon’s surface;

4. the mass of a man whose weight has a magnitude of 700N on the Earth’s surface.

Is this mass the same over the Moon’s surface?

[0,91N; 14N; 212N; 71,4kg ]

• Exercise 8. (In your exercise book) True or false? Justify your answer.

1. The direction of weight is always vertical.

2. The weight of a body can be represented by an arrow applied to its center of gravity.

3. The magnitude of weight can be expressed in kilograms.

4. When a spacecraft leaves the Earth, the magnitude of the gravitational attraction of the Earth on it decreases in such a way that if its distance from the Earth’s center doubles, then the magnitude of the gravitational attraction halves,

5. The force exerted by the Earth on the Moon is more intense than the force exerted by the Moon on the Earth.

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• Exercise 9. Complete the following table :

The planet Venus has a mass mv = 4,88·10²⁴kg. Its diameter isdv = 1,225·10⁴km.

a) Using these data, calculate the acceleration of the gravity g at the surface of Venus, then check your result with a CRM table.

An astronaut weighs himself using terrestrial scales indicating a (false) value of 32kg on Venus’ surface.

b) What is the weight acting on the astronaut?

c) What is the true mass of the astronaut?

d) What are his mass and his weight on the Earth’s surface, in Geneva?

[ a)8,68_kg^N; c)36,2kg ]

We recall that, near the Earth’s surface, the value of g changes as a function of the height above the ground.

a) If g = 9,81_kg^N at ground level, what is its value at 10 000m, the altitude at which a passenger airplane flies?

b) According to you, is the choice of flying at this altitude a consequence of the variation of the value ofg?

[ 9,78_kg^N ]

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Star Trappist-1 is among the closest to the Sun, only few tenth of years-light, and owns seven temperate exoplanets similar to the Earth, the largest number ever observed (in 2017). The first of those seven planets, called “exoplanet B”, has a mass of 0,85 times the Earth’s mass, and a radius of 1,09 times the Earth’s radius.

Its distance from the star Trappist-1 is 1,66· 10⁹m, and the gravity’s force of Trappist-1 on exoplanet B is2,0·10²⁵N intense.

Vision d’artiste de Trappist-1.

a) Draw a sketch representing star Trappist-1 and exoplanet B, indicate all the needed quantities with a symbol (a letter).

b) On the same sketch, draw the force of Trappist-1 on exoplanet B, and the force of exoplanet B on Trappist-1, with an appropriate scale.

c) Calculate the mass of star Trappist-1.

d) Compare the mass of Trappist-1 to the mass of the Sun by calculating their ratio.

e) Calculate the gravity’s acceleration on the surface of exoplanet B:g_B. Then compare it to the Earth’s one by their ratio.

[1,6·10²⁹kg; 8,0%; 7,02N/kg; 71,6% ]

• Exercise 13. (In your exercise book) A neutron star is the collapsed core of a large star. It is an astrophysical object extremely compact: the smallest neutron stars have a mass Mns = 1,4MSun, though their typical radius is only 10km long.

a) Using the data given above, calculate the gravity’s acceleration on the surface of a neutron star, g_ns. Then compare it to g_Sun by calculating their ratio.

b) If a man of 80kg stood on the surface of a neutron star, what would his weight be?

c) Which value for his mass would a terrestrial scale indicate (if a strong enough scale existed)?

[ 1,9·10¹²N/kg; 6,9·10⁹; 1,5·10¹⁴N; 1,5·10¹³kg ]

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3 Net force and equilibrium of a material point

• Exercise 1. For each case, draw the net (or resultant) force with a red pen. Then find the magnitude of each force and of the net force. Scale: 1cm ←→5N.

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• Exercise 2.

a) Draw the net resultant force with a red pen, then find the magnitude of each force and of the net force. Scale: 1cm ←→50N.

b) Draw the force F~₀ such that the sum of all the forces is zero in each case. Then find the magnitude of all the drawn forces. Scale: 1cm ←→50N.

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c) DecomposeF~_rs into a couple of forces F~₁ and F~₂ along the indicated directions.

d) Draw two forces F~₁ and F~₂ along the indicated directions, such that the sum of all the forces is zero in each case. Then find the magnitude of all the drawn forces.

Scale: 1cm ←→50N.

F0 F0

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• Exercise 3. (In your exercise book) A mass of 450g is hung with a wire.

1. Draw the system-interaction diagram for the mass.

2. What is the weight of the mass?

3. Draw the diagram of the forces acting on the mass, using the scale 1cm←→1N.

4. What is the net force acting on the mass?

5. What is the force of the wire on the mass?

(direction, way and magnitude)

An upward force of intensityF = 80Nis required for keeping a suitcase lifted (at rest).

a) Draw the system-interaction diagram and the force diagram for the suitcase, using the scale 1cm←→20N.

b) What is the weight of the suitcase?

c) What is the mass of the suitcase?

Consider the magnet of mass m = 200g in levitation above a magnetic slab shown in the following figure.

1) Draw the system-interaction diagram of the situation.

2) Draw the forces acting on the magnet, indicating the chosen scale.

3) What is the net force on the levitating magnet?

4) What is the magnitude of the push-up force between the magnet and the slab?

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A book of massm_b = 400g is put on a table of massm_t= 8,00kg.

a) Draw the system-interaction diagram of the situation.

b) Draw 2 forces diagrams: one for the book and one for the table (define a scale).

c) What is the magnitude of the force of the table on the book?

d) What is the magnitude of the force of the ground on the table?

[ 82,4N ]

A packet of mass m = 500g is hung from the ceiling using a wire whose mass is negligible.

Someone pulls the packet downwards with a force of magnitude 10N. What is the magnitude of the force of the wire on the ceiling? (Draw the system-interaction diagram of the situation and the forces diagrams for the packet and for the wire.)

[15N ]

• Exercise 8. A pendant has a mass of 200g and is at rest, hung on a chain as shown in the figure below.

Draw the diagram of the three forces acting on the pendant. (Use the scale1cm←→1N.) Determine the magnitude of the two forces of the chain on the pendant.

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• Exercise 9. .

For both the following situations, the lifted bowl is at rest, and its mass is8,00kg.

1. Draw the vector representing the weight of the bowl on the diagram of situation I. For simplicity, choose the point where the strings join as the application point.

Use the scale 1cm←→20N.

2. In the same diagram, draw the forces that the two men have to apply to achieve the equilibrium.

3. Repeat the points 1) and 2) for the diagram of situation II.

4. What is the difference between the magnitudes of the forces in the two situations?

5. How can you explain this difference, knowing that the mass to be lifted is the same? What is the additional force required in the situation II?

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• Exercise 10.

The weight of the cable car in the following figure is 8000N. It is at rest.

1. Draw the weight force of the cable car. For simplicity, apply this force at the hanging point and use the scale 1cm←→2000N.

2. Draw the tensions on both sides of the cable which are acting on the cable car (by geometrical construction) and find their magnitude using the scale.

[ 14000N; 9600N ]

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• Exercise 11.

A trolley has a weight 0,900N intense, and it is hold motionless by a string on an inclined plane.

1. On the figure below, draw the weight of the trolley. Choose a scale such that its length is at least 3cm.

2. By geometrical construction, draw all the other forces acting on the trolley.

3. Determinate the intensity of each force using the scale and using the right angle trigonometry.

For simplicity, choose the pointG as the application point for all the forces.

[ 0,380N; 0,816N]

25°

G

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A bowl of mass m = 7,50kg is held motionless by a string and a wall, as shown in the figure below.

1. Draw the system-interaction diagram for the bowl.

2. Draw all the forces acting on the bowl, applied in its center for simplicity. Use the scale 1cm←→20Nand name each force.

3. Measure the angle between the string and the vertical wall.

4. What is the magnitude of the tension of the string on the bowl? Find it both graphically (using the scale) and mathematically (using right-angled triangle trigonometry).

[ 81,2N ]

• Exercise 13.

A bowl of 20,0kg and 20,0cm radius is lying over a plane inclined at 20,0^◦, held at rest by a vertical wall.

1. Draw a system-interaction diagram for the bowl.

2. Draw all the forces acting on the bowl, applied in its center for simplicity. Use the scale1cm ←→50Nand name each force.

3. Find the magnitude of each force both graphically (using the scale) and mathematically (using right-angled triangle trigonometry).

[ 196N; 71,4N; 209N ]

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• Exercise 14. Vaiana has a mass of 50kg, and she is at the equilibrium on the boat.

1. Calculate the intensity of the weight of Vaiana, then draw this force with the scale 1cm←→100N.Choose the center of mass of the girl (indicated) as the application point.

2. Using the same scale and the same application point, draw the other two forces acting on Vaiana, whose directions are indicated with two dashed lines: the force of the rope, F~₁, and the force of the boat , F~₂. Précision needed. Use a visible colour.

3. Find the intensities of F~₁ and of F~₂.

4. The mass of Maui is120kg. Knowing that the rope can support the maximum tension of1800N, would the rope break if Maui was in place of Vaiana?

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• Exercise 15. A basket ball of massm= 855g is at rest, fallen between a vertical wall and the roof of a cottage forming an angle α = 25^◦ with the horizontal line, as shown in the figure.

1. Draw the system-interaction diagram for the ball.

2. Draw the forces diagram for the ball, choosing its center as the application point.

Use the scale 1cm←→2N and name each force.

3. Find the magnitude of each force both graphically (using the scale) and mathematically (using right-angled triangle trigonometry).

[ 9,3N; 3,9N ]

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• Exercise 16. A dynamometer pulls a piece of wood horizontally, on a table. The mass of the wood is m = 1,2kg. The static friction coefficient between the wood and the table is µ₀ = 0,60. The dynamic friction coefficient is µ = 0,40. What does the dynamometer indicate

1. before the piece of wood moves?

2. at the very instant the piece of wood starts sliding on the table?

3. while the piece of wood is sliding at a constant velocity?

[ <7,1N; 7,1N; 4,7N ]

• Exercise 17 (In your exercise book). Caty and Julian are trying to move a closet having a mass m = 80kg from one side of their bedroom to the other. Caty pulls the closet using a cable; she pulls horizontally with a force of magnitude F_C = 80N. Juilan pushes the closet from behind. He applies a horizontal force of magnitude F_J = 120N.

Despite all their labors the closet doesn’t move.

1. Draw system-interaction diagram for the closet.

2. List all the forces acting on the closet and, for each of them, give its features (direction, way and magnitude).

3. Draw the forces diagram for the closet using the scale 1cm←→100N.

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4 Pressure

4.1 Pressure between solids

• Exercise 1. The brick in the figure below has three different faces. On which face should it be put to have the greatest pressure on its base? Justify your answer.

1. Is pressure proportional to the surface for a fixed force acting perpendicularly to the surface? Justify your answer.

2. In the adjacent graphic, which curve represents the pressure as a function of the surface for a fixed force? Justify your answer.

3. Is pressure proportional to the magnitude of the force acting perpendicularly to a given surface? Justify your answer.

4. In the adjacent graphic, which curve represents the pressure as a function of the magnitude of the force for a fixed surface? Justify your answer.

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•Exercise 3. A cube made of iron whose sides measure10,0 cmis put on a table. Find the pressure between this cube and the table.

[ 77,1·10²Pa ]

The average pressure between the tires of a car having a mass m=1,0t and the ground is 2,0·10⁵Pa. What is the area of the contact surface between each tire and the ground?

[ 120cm² ]

The bridge in the figure below is supported by a couple of pillars which sink into the ground if the pressure exceeds 80kPa. The magnitude of the bridge’s weight isF_g = 1,2 megaNewtons (MN). What is the minimal surface that each pillar must have to avoid the bridge sinking?

[ 7,5m² ]

The maximum pressure allowed on a floor is1,5·10⁷Pa. Convert this value intoN/cm². How many students of mass 50kg can get onto a table whose four feet are cylinders having a diameter of 2,0cm, without breaking the floor? (Neglect the weight of the table.)

[ 1,5·10^{3 N}_cm2; 38 students ]

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• Exercise 7. The picture below shows a physicist lying on a bed of nails. What is he trying to show? (Don’t try to do it yourself!)

1. A pressure greater than 4,5N/mm² on your skin starts hurting. Let us suppose that the point of each nail has a surface of 1,5 mm². Calculate the weight that this man may exert on each nail without feeling pain.

2. If the weight of the physicist is 690N, what is the minimum number of nails that must be there for him not to feel pain?

3. Explain why he must be careful when lying down or in getting off the bed of nails.

[ 1) 6,75N; 2) 103 nails ]

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An ice skater stands on one foot. The blade of the skate is 2,0mm thick and 23,2cm width. Calculate the pressure between the ice and the blade if the mass of the skater is 60kg.

[ 13bar ]

A steel ruler measures 32,0cm and has a square section whose side is 8,00mm long.

Calculate the pressure under this ruler

- when it lies horizontally on one of its rectangular faces;

- when it lies vertically on a square face.

[ 616Pa; 24600Pa ]

Pressing on a pin with a force of magnitude 10,0N, what is the pressure on the pin if the diameter of its point is 0,0050mm?

[5,1·10¹¹Pa ]

• Exercise 11.

The edge of the second cube is double that of the first cube and both cubes are made of the same substance. What can you say about the pressure on the bottom of the second cube with respect the first one?

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Two cylinders have the same mass of 540g and are made of the same metal. Both stand on a table. The height of the first cylinder is 12,0cm and its pressure on the table is 31,8hPa. The height of the second cylinder is 8,00cm.

a) Calculate the radius of the first cylinder. Give your answer in cm.

b) Calculate the pressure between the second cylinder and the table.

[ 2,3cm; 21,3hPa ]

• Exercise 13. The figure below shows Boris the cat, Vladimir the kitten and Leonid the lion. Boris is twice as tall, twice as wide and twice as long as Vladimir, and the felines have the same density.

1. Compare the volumes of Vladimir and Boris. What can you say about their weights?

2. The paw print (the surface of the paw which is in contact with the ground) of Boris is bigger than Vladimir’s. How much bigger, in your opinion?

3. Is the pressure greater under the paw of Boris or of Vladimir? What is the ratio between those two quantities?

4. Leonid is a distant cousin of Boris, and he is much bigger. Use the result of the previous question in order to explain qualitatively the reason why lion’s paws are thicker than cat’s paws.

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4.2 Pressure in a fluid

a) Using a needle you can suck up a liquid in a container, and this by pulling a piston while the other end is immersed. Explain the reason why the liquid goes up in the needle.

b) In the 17^th century, the hydrant men of Florence wanted to suck up the water in the river Arno to supply the fountains of the town. They used a pump working on the same principle as a needle and observed that water did not rise more than a certain height.

Calculate this height.

c) What would the maximal height have been if they had sucked up mercury rather than water?

• Exercise 15.

Classify from the weakest to the highest the pressures at points A, B, C, D and E of the container full of liquid shown in the adjacent figure. Does the liquid exert a force on the sides of the container at points C and D? If yes, draw the respective forces, with the right direction (not to scale).

The bottom of a boat is 30,0cm below the surface of the sea (ρ = 1030_m^kg3, P_atm = 1,00 · 10⁵Pa) and has a hole with a circular section whose area measures 6,00cm². Calculate

a) the total pressure at the hole’s depth;

b) the magnitude of the pressing force of the water on the plug closing the hole.

c) the magnitude of the force that has to be applied on this plug to close the hole (neglect the plug’s weight).

[ 1,03·10⁵Pa; 61,8N; 1,82N ]

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A nuclear submarine is diving 300m under the surface of the sea. The pressure of the air inside the submarine is of 980hPa, while the atmospheric pressure is of 1030hPa.

Calculate the pressing force that must act on a rectangular door measuring 80cm times 60cm, located on the bottom of the submarine, in order to keep this door close.

What mass would have a weight of the same magnitude?

[ 1,5·10⁶N;150tons ]

A U-shaped tube contains some water. 10cm of oil are poured into one of its branches (oil does not mix with water). Is the top level of the the oil lower, higher or the same as the top level of the water in the other branch? If the levels are different, calculate the difference between them.

[ 1,6cm ]

A U-shaped tube of glass partially filled with water allows one to measure the pressure of a gas delivered by a pipe. The U tube open at both ends, is first placed in the ambient air.

1. In the figure below, draw the water level in the right branch of the tube. Justify your answer.

2. A rubber tube joined to the gas pipe is fixed to the left branch of the tube. The water is partially pushed toward the right branch. In such conditions, the difference in the height of the the levels of the water in the right and left side is h=60cm.

Calculate the excess pressure of the gas with respect to the atmospheric pressure.

3. Calculate the pressure of the gas if the barometer in the laboratory indicate an atmospheric pressure of 95kPa.

[ 2) 6,2%de P_atm; 3) 1,01bar ]

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• Exercise 20. (In your exercise book) Referring to the adjacent figure, h_A = 3,2m and h_B = 5,7m. Cal- culate the pressure at the surface of the water inside the submarine cav- ity (point A). The atmospheric pressure is 1020hPa.

[ 1,3·10⁵Pa ]

An hydraulic press is formed by two vertical cylinders filled with oil. In the little cylinder on the left side of the press, whose radius is r = 1,25cm, the oil is in contact with the little pistoni. The bigger cylinder on the right side has a radiusR = 3,75cm. Here the oil is in contact with the big piston I. Both pistons are initially in equilibrium at the same height, as shown in the figure below.

1. Pushing downward onto the little piston with a force of magnitude F₁ = 30N, the big piston is forced upward. Show that the force on the big pistonF₂ (upward) is more intense than the force acting downward on the small piston F1.

2. Calculate F2.

3. We want to lift a load of mass M = 50kg lying on the big piston, as shown in the figure below, on the right. What minimal mass m should we put onto the little piston for that?

[ 2) 270N; 3) ≥5,6kg ]

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5 Archimedes’ principle

• Exercise 1. (In your exercise book) True or false? Justify your answer.

1. Archimedes’ buoyant force on a body only depends on the nature of the liquid.

2. Archimedes’ force acts always vertically and upward.

3. The greater is the volume of a body, the more intense is Archimedes’ force on this body when immersed.

4. Archimedes’ force on a body can never be more intense than the weight of this body.

5. There exists at least one liquid on which a steel bolt floats.

• Exercise 2. Choose the right answer. Justify your choice and explain why the other answers are wrong (remember that a counterexample is enough!).

A balloon inflated with helium rises in the air because:

1. at the same pressure, helium is lighter than air;

2. the pressure of the helium creates many pressing forces on the inside surface of the balloon, whose resultant force is upward and more intense than the weight of the balloon;

3. the pressure of the air creates many pressing forces on the outside of the balloon whose resultant force is upward and more intense than the weight of the balloon;

4. the balloon is sucked up by the vacuum prevailing above the atmosphere.

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• Exercise 3. (In your exercise book) In the following figure, the volume of the object made of aluminum is three times bigger than the volume of the other object, made of iron. When both are immersed, the scales

1. stay balanced;

2. tilt toward the aluminum side;

3. tilt toward the iron side.

Justify your choice by calculation.

• Exercise 4. (In your exercise book) In the following figure, the two objects hung on the scales are identical. When they are immersed, one object in the alcohol and the other in the water, the scales

1. stay balanced;

2. tilt toward the water side;

3. tilt toward the alcohol side.

Justify your choice.

• Exercise 5. (In your exercise book) A boat on a lake has a mass of 2,5 tons. What is the mass of water that it moves?

[ 2,5m³ ]

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A piece of iron has a mass of 780g and might have some holes inside. When immersed in water, the decrease of its weight is of 1,53N. We want to know whether there are actually any holes and, if this is the case, determine their total volume.

[ Yes, 56,3cm³ ]

• Exercise 7. (In your exercise book) A block made of wood (ρwood = 450_m^kg3) floats in the water, as shown in the adjacent figure. Cal- culateh, the immersed height.

[ 4,5cm ]

A weather balloon has a mass of 5,00kg when it is empty, and it has a radius of 2,879m when it is entirely inflated with helium. It bears a load with a mass of 10kg. Can the balloon take off with this load? Justify by computation.

[ Yes ]

Calculate the minimum volume of an aerostat inflated with helium to be able to carry a total load of 655kg including: the passengers, the basket, the rigging and the envelope.

[ 589m³ ]

A hot-air balloon has a volume of 2950m³. Knowing that the density of the hot air inside the balloon is80%of the density of the cooler air outside, calculate the maximum load (passengers, basket, rigging, envelope) that the balloon can carry. Useρair outside = 1,29_m^kg3.

[761kg ]

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An ice cube floats in a glass full to the top of water. When the ice melts, does the water overflow?

A hollow sphere has an inside radius r = 9,0cm and an outside radius R = 10cm. The sphere is half-immersed in a liquid whose density is80% of the density of water.

- Find the density of the material of the sphere.

- What should be the density of the liquid for the sphere to float completely immersed?

A cylindrical buoy with a diameterd = 80cmand mass M = 1400kg floats vertically in salt water (whose density is1030_m^kg3).

- What fraction of the buoy is immersed if a girl of 60kg climbs onto it?

- What is the period of the vertical oscillations when the girl dives?

A block of lead (ρ_lead = 11300_m^kg3) is set over a piece of wood of mass m = 180g (ρ_wood = 600_m^kg3). What should be the mass of the block of lead for the wood to float in a liquid whose density is 900_m^kg3, with ⁴₅ of its volume immersed?

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6 Turning effect of forces

• Exercise 1.

Calculate the moment of the force applied to the nut using the adjustable spanner, knowing that the force represented is on a scale of 1cm←→50N.

• Exercise 2.

Complete, in the case of rotational equilibrium:

F1 = 20N F2 = 30N d1 = 6m d2 =...;

F₁ = 50N F₂ =... d₁ = 20m d₂ = 5cm;

F₁ =... F₂ = 100N d₁ = 4m d₂ = 2m;

F₁ = 50N F₂ = 5N d₁ =... d₂ = 50cm.

(1)

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• Exercise 3.

Complete, in the case of rotational equilibrium:

F₁ = 40N F₂ = 30N F₃ = 40N d₁ = 7m d₂ =... d₃ = 4m;

F1 = 20N F2 = 30N F3 =... d1 = 15m d2 = 30m d3 = 6m.

(2)

1. Is the see-saw in the figure below balanced? If not, which side tips down? Justify your answer.

2. If the see-saw is not balanced, where should Albert (whose mass is 12kg) sit in order to have equilibrium?

Masses: Paul→34kg, Anne →46kg, André→20kg, Luc →30kg et Lucy→25kg.

[ 1,77m on the right ]

(37)

• Exercise 5. (In your exercise book) Mr. Norbert Lingot is carrying some sand in a wheelbarrow to build a bar- becue in his garden.

a) Draw all the forces active when he lifts the handle vertically.

b) Is the wheelbarrow a one-arm lever or a two-arm lever? Justify your answer.

c) The wheelbarrow, whose mass is20kg, contains60kgof sand. Let us call G the center of mass of the system (wheelbarrow + sand). Find the magnitudeF of the vertical force that Norbert has to apply on the wheelbarrow to lift it.

d) To pour the sand, Norbert must tilt the wheelbarrow. Should he keep the direction of his force vertical? Justify your answer drawing a sketch.

[ c) 240N]

• Exercise 6.

The crowbar in the adjacent figure is used for pulling out a nail. Why use such a tool for this kind of task? Draw the pivot and the moment arm on the adjacent figure.

(38)

A load of mass M = 1500kg is fixed to the cable of a crane.

Knowing thatOA = 3,0m and OB = 10m, 1. calculate the moment of the weight of

the load M with respect to the pivot O.

2. What must be the mass of the counter- weight Mc for the crane to be in rotational equilibrium?

[ 1,5·10⁵N; 5000kg ]

• Exercise 8. The adjacent figure shows a skinned human arm.

1) Draw on this picture (without warning about scale) the weight of the object and the pulling force of the muscle at the point A (where the tie is attached to the forearm).

2) Indicate the pivot, and the momentum arm.

3) Underline the right answer:

the magnitude of the pulling force of the muscle on the forearm to maintain rotational equilibrium is

half - equal to - double - quadruple the magnitude of the weight of the object.

(39)

Monsieur Labricole carries 50 tiles in a wheelbarrow. Each tile has a mass of 800 g, and the wheelbarrow has a mass of 10kg.

a) Find the total mass that M. Labricole has to lift (wheelbarrow + tiles).

b) What is the magnitude of the weight corresponding to this mass?

The loaded wheelbarrow is in equilibrium, and it is subject to three forces:

1. the force F~ exerted by M. Labricole at point A, vertically and upwards;

2. the reaction of the groundR,~ applied at pointCvertically and upward, and passing through the rotation axisO;

3. the total weight P~ (calculated under b) above), with application point G, the center of mass of the system.

c) DrawP~ in the picture above. Use a scale of 1cm←→200N.

d) Assuming the rotational equilibrium, calculate the momentum of the force F~ with respect to the pivotO (let us call it M_F).

e) DeduceF, then draw F~ in the picture above.

f) Can you find the magnitude ofR?~ g) Complete the following table:

[ b) 490N; d) 250N·m; e) 196N; f) 294N ]

(40)

• Exercise 10. (In your exercise book) The mass of the wheelbarrow with Mme Bidochon is80kg.

What is the force exerted by M. Bidochon to raise this wheelbarrow?

[ 42N]

• Exercise 11. (In your exercise book) To paint the front of a house, a worker stands in a cradle, which is suspended by a couple of cables fixed at points A and B, as shown in the figure. The mass of the system “worker + cradle” is 300kg, and it is in mechanical equilibrium and undergoes the action of three forces:

- the weight of the system “worker + cradle” P~;

- the tension of the cable passing through A and C, say F~A;

- the tension of the cable passing through A and C, say F~_B.

1. Find the magnitude of the weight P;

2. calculate the momentum of P~ with respect to point A (which we consider as a possible rotation axis): M_P;

3. calculateF_B knowing that the system “worker + cradle” is in rotational equilibrium with respect to the point A;

4. calculateF_A knowing that the system “worker + cradle” is at rest (in translational equilibrium).

[ 1) 2940N; 2) 8090N·m; 3) 1620N; 4) 1320N ]

(41)

A beam of mass m= 50g is hung on a dynamometer, and one of its extremities lies on the edge of a table. Calculate the magnitude of the force exerted by the dynamometer in situations a) and b) of the figure below:

[ a) 0,25N; b) 0,42N ]

• Exercise 13. (In your exercise book) The adjacent figure shows a steelyard.

This is a hanging two-arm balance, with a pivot at pointC. The package fixed to point A has a mass m₁ = 1,6kg. The reference mass is m₂ = 200g.

The distance between the two masses isd= 70cm.

- Calculate the distance d₂ when the scale is in equilibrium (neglecting the arm’s weight).

- How could this instrument be used to determine the unknown masses of other packets?

[ 62cm ]

(42)

The “capstan” is a piece of machinery developed for mariners in the past, when the engines did not exist. It was useful for doing jobs that requires great strength.

To understand how a capstan works, let us consider one having a winding of diameterd= 0,50m,and four sailors pushing with the same intensity at a distanceLof1,3mfrom the pivot.

What is the minimal force that each sailor must supply to raise a sail ex- erting a force of10⁴Non the winding

rope? [ 5·10²N ]

• Exercise 15.

The drawbridge of the castle in the adjacent figure has a mass of 5,00 tons. Knowing that α= 30^◦, find what the minimum magnitude of the tension of the chainF~ must be to raise the drawbridge.

[ 49·10³N ]

Physics exercises for the 1DF course