Name : Class : A2 PhysicsSubject : A2 Physics Lesson No : 1Chapter : 8 Gravitational Field Date : 10/12/13 (Tue)

Time : 10.00pm-12.00pmTopic : 8.1 Gravitational field, Newton’s law of

gravitation, & gravitational field strength

A2 [18]Chapter Topic7 Motion in a Circle8 Gravitational Field11 Ideal Gases12 Temperature13 Thermal Properties of Materials14 Oscillations17 (AS) Electric Fields18 Capacitance21 Magnetic Fields22 Electromagnetism23 Electromagnetic Induction24 Alternating Currents25 Charged Particles26 Quantum Physics27 (AS) Nuclear Physics28 Direct Sensing29 Remote Sensing30 Communication Information

Scheme of Assessment

PAPER TYPE DURATION MARKS WEIGHTINGAS Level A Level

1 MCQ 1 hr 40 31 % 15 %2 Structured 1 hr 60 46 % 23 %3 Practical 2 hr 40 23 % 12 %4 Structured (12Qs)[8+4] 1 hr 45 min 100 38 %5 Planning, Analysis and

Evaluation (2Qs)1 hr 15 min 30 12 %

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Newton's Law of Gravitation

1. Newton's law of universal gravitation is stated as follows:A point mass will attract another point mass in any part of the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

2. Mathematically where F is the gravitational force,m1 is the mass of particle 1,m2 is the mass of particle 2.r is the distance between particle 1 and 2,G is the universal gravitational constant.

3. This law is applicable only between point masses. Every spherical object with constant density can be considered as a point mass at the centre of the sphere.

4. The gravitational forces between two point masses are equal and opposite and constitute an action and reaction pair of forces (Fig 7.1). They are attractive in nature and always act along the line joining the two point masses.

5. G is known as the universal gravitational constant and has a value of 6.67 x 10-11 N m2 kg-2. It is not affected by the type of substance . Temperature has no effect on its value. It is therefore a constant throughout the universe.

6. Since G is very small, gravitational forces become significant only when we are dealing with massive bodies. Gravitational forces between atoms and molecules are insignificant.

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7. Sign conventions(a) The positive direction is taken as the direction of increasing distance

from a fixed point (Fig 7.2). In the case of the Earth, any displacement away from the Earth's centre is positive.

(b) The gravitational force acting on the Moon due to the Earth is directed towards the Earth's centre. This force is attractive and is written as:

. The negative sign is ignored when only the magnitude of the force is required.

A Levels Practice 11. The Earth may be considered to be a sphere of mass 6.0 x 1024 kg and radius

6.38 106 m. Calculate the gravitational force acting on a(a) 1.0 kg mass on the Earth's surface,(b) 1.0 kg mass placed at a distance of 3.8 x 108 m from the centre of the Earth.(c) Calculate the initial acceleration of the 1.0 kg mass if it is released from its position of rest in (b).

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2. The mass of the Earth is 5.98 x 1024 kg and that of the Moon is 7.35x 1022 kg. A spacecraft travelling from the Earth to the Moon will reach a point X where it experiences no resultant gravitational force. The distance between the centre of the Earth and the centre of the Moon is 3.85x 107 m.

(a) Draw a free-body diagram of the spacecraft at X.(b) Calculate the distance from X to the centre of the Moon.(c) Describe the motion of the spacecraft before reaching X and after leaving X.

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Gravitational Field of a Point Mass

1. A gravitational field is said to exist at a point if a gravitational force F is exerted on a test mass in placed at that point in space.

2. The gravitational field strength g at a point is defined as

3. This is a vector quantity.4. Gravitational field strength is also known as gravitational acceleration or the

free-fall acceleration. Its S.I. unit is N kg-1. It can also be expressed in S.I. base units of m s-2.

5. Another formula for the gravitational field strength at a point can be obtained by considering the point mass m placed at a distance r from another point mass M. The gravitational force acting on m is

6. The gravitational field strength at a distance r from the mass M is

Variation of g, Acceleration due to Gravity with Distance r from the Centre of the Earth

Explaination:

1. For a mass m on the surface of the Earth

2. Acceleration due to gravity on the surface of the Earth, g = G

3. For r > R, acceleration due to gravity

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4. Figure below shows a mass m at a distance r < R from the centre of the Earth.

The part of the Earth that attracts the mass m is of radius r and mass

, The whole mass of the Earth,

5. If g' = acceleration due to gravity at a distance r from the centre of the Earth, (r < R),

6. From the equation,

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A Levels Practice 21.

Fig 7.4 shows a mass m resting on the surface of the Earth of mass M. Take the acceleration of free fall near the Earth's surface to be 9.81 mAssume the Earth to be a uniform sphere of radius 6.37 x 106 m.(a) Draw the gravitational force which

(i) M exerts on m. Label it as Fm.(ii) m exerts on M. Label it as FM.

(b) Calculate the mass of the Earth.

(c) How far from the Earth's surface must the mass m be moved in order for the gravitational field strength to be reduced to half that on the Earth's surface?

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2. The mean density of the Moon is that of the Earth. The acceleration of free

fall on the surface of the Moon is times that on the surface of the Earth. Find the ratio of the radius of the Earth to the radius of the Moon.

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3. The earth may be assumed to be spherical with radius rand density . Which equation correctly relates the gravitational field strength g at its surface to these quantities and the gravitational constant G? C

A. C.

B. D.

4. When a satellite is at its launch site on the Earth's surface, it is found to have a weight W. When the satellite is placed in a circular orbit at a height h = 6R above the Earth's surface, where R is the radius of the Earth, what is the gravitational force acting on it? D

A. W/6 C. W/36B. W/7 D. W/49

5. A uniform sphere of diameter d has a gravitational field strength X at its surface. A seconduniform sphere has the same density but has a diameter 2d. What is the gravitational field

strength at the surface of the second sphere? A

A. 2X B. 4XC. 8X D. 16X

6. At the Earth's surface, the gravitational field strength is about 10 N kg -1. At a point outside the Earth and a distance x from its centre, the field strength is about 5 N kg-1. Which expression gives an approximate value for the radius of the Earth? D

A. C.

B. D.

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Challenging Questions

1. Which statement about a uniform gravitational field is true? D

A. Its magnitude is the same in all directions.B. The gravitational potential has the same value at all points within it.C. Its direction is opposite to the direction of motion of a test mass released in it.D. Its field strength is the same at all points within it.

2. Star X of mass 2M and star Y of mass M are moving in circular orbits around their common centre of mass under their mutual gravitational attraction. The effects of any

other bodies may be ignored. What is the value of ? C

A. 4 B. 2 C. 1 D. 1/2

3. [NOV07/4/2b] (b) Calculate

(i) the gravitational force between two Helium-4 atoms that are separated by a distance of 3.4 x 10-9 m, [3]

(ii) the ratio

4. (i) The metre was first defined as one ten-millionth of the distance from the equator to the north pole along the earth's surface. Using this information, estimate the radius of the earth. Write down one assumption made in your estimation. [2]

6.37 x 106 m. Earth is a uniform sphere.

(ii) Using Newton's law or gravitation and your answers to (i), find the gravitational force acting on a l.0 kg mass at the surface of the earth. The earth may be assumed to be a sphere of mass 6.0 x 1024 kg. 9.9 N

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5. Both the Sun of mass MS, and the Moon of mass Mm, exerts gravitational forces on the Earth. The distance of the Sun from the Earth is rS, and the distance of the Moon from the

Earth is rm. Which expression gives the value of ? D

A. C.

B. D.

6. The given diagram shows two stars of equal mass M moving at constant speed v, in a circular orbit of radius R around their common centre of mass. Which expression gives the resultant force on each star? A

A. C. zero

B. D.

7. The Earth may be taken to be a uniform sphere rotating about an axis through the poles. Which of the following correctly compares the weight of a body at the Equator relative to its weight at a pole? D

A. greater, because angular velocity of the Earth is greater at the Equator than at a pole.

B. greater, because the weight at the Equator is given by the sum of the gravitational attraction of the Earth and the centripetal force due to the circular motion of the body.

C. the same, because the weight is the gravitational attraction of the Earth and for a uniform sphere, even when rotating, this is independent of the body's position on the Earth.

D. smaller, because the gravitational attraction of the Earth must provide both the weight and the centripetal force due to the circular motion of the body.

8. A planet of mass M and radius r is rotating so rapidly that objects at its equator only just remains on its surface. In the expressions below, G is the gravitational constant. Which expression gives the planet’s period of rotation? C

A. C.

B. D.

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9. [JUN09/4/la] (a) Define gravitational field strength. [1]

(b) Draw the graph of gravitational field strength, g against distance from centre of the earth, r.

10. A body of mass 20kg is placed 4m above the earth's surface. The value of g may be taken as 10ms-2 . What are the gravitational field strength and gravitational force acting on the body? C

Gravitational field strength / N kg-1 Gravitational force / NA.B.C.D.

0.5101040

1010200200

11. [JUN09/4/lb] (b) A spherical planet has diameter 1.2 x 104 km. The gravitational field strength at the surface of the planet is 8.6 N kg -1. The planet may be assumed to be isolated in space and to have its mass concentrated at its centre. Calculate the mass of the planet. [3]

M = 4.6 x 1024 kg

12. At the surface of a planet the acceleration of free fall is greater than that on Earth. If the planet has the same radius as the Earth, which of the following on the planet will be the same as that on the Earth? D

A. the escape velocity of a spacecraft.B. the weight of an astronaut as measured by a spring balance.C. the height to which an astronaut can jump.D. the surface tension of a liquid.

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13. The acceleration of free fall g on the surfaces of two planets are the same. Which of the following expressions must be the same for the planets? D

A. mass B. radiusC. mass / radius D. mass / (radius)2

14. The gravitational field strength outside a uniform sphere of mass M is the same as that of a point mass M at the centre of the sphere. The Earth, which may be assumed to be spherical and of radius r, has a gravitational field strength g at its surface. Which expression gives the gravitational field strength at height h above the ground? A

A. C.

B. D.

15. (a) The Earth may be assumed to be a uniform sphere of radius 6370 km, spinning on its axis with a period of 24.0 hours. The gravitational field at the Earth's surface is identical with that of a point mass of 5.98 x 1024 kg situated at its centre. For a 1.00 kg mass located at the Equator, [6]

(i) use Newton's law of Gravitation to determine the gravitational force on the mass,

9.83 N

(ii) calculate the force required to maintain the circular path of the mass,

0.0337 N

(iii) determine the reading on an accurate newton-meter (spring balance) supporting the mass.

9.80 N

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(b) From your answers to (a), write down what would be the acceleration of the mass at the Earth's surface due to [2]

(i) the gravitational force alone, 9.83 m s-2

(ii) the force as measured on the newton-meter. 9.80 m s-2

(c) At the Equator, a ball is released from rest in a vacuum and its acceleration towards the Earth's surface is measured. The acceleration is then stated as 'the acceleration due to Gravity'. Comment on the statement.

Incorrect.Measured acceleration is less than the actual acceleration due to gravity. Part of g0 is used to maintain rotational motion.

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