UNIT 7MOTION AND FORCE
Portrait of Isaac Newton (1642-1727) |
Exercise
1. Choose the correct option for the following questions:
(a) What is the relation between the distance between two objects (d) and the gravitational force (F) produced between them?
(b) What is the change in the gravitational force between two objects when their mass is doubled?
(c) If the gravitational force between two objects on Earth is 60 N. what is the gravitational force between those two objects on the moon?
(d) Which one of the following statements is correct?
(e) At which of the following places do you weigh the most?
(f) The radius of the Earth is 6371 km and the weight of an object on the earth is 800 N. What is the weight of the object at a height of 6371 km from the surface of the earth?
(g) If the mass and the radius of a celestial body are two times the mass and the radius of the earth respectively, what is the value of acceleration due to the gravity of that body?
(h) What will be the weight of a man on the moon, if his weight on earth is 750 N? (The acceleration due to the gravity of the moon = 1.63 m/s² )
(i) The mass of planet B is twice the mass of planet A but its radius is half of the radius of planet A. Similarly, the mass of planet C is half of the mass of planet A. but its radius is twice the radius of planet A. If the weight of an object in planets A, B and C is W1, W2 and W3 respectively, which of the following order is correct?
(j) Which one of the following conclusions is correct while observing a freely falling object every second?
2. Differentiate between:
(a) Gravitational constant G and acceleration due to gravity g
Difference Between Gravitational Constant (G) and Acceleration Due to Gravity (g)
Gravitational Constant (G)
Acceleration Due to Gravity (g)
1. The gravitational constant (G) is a universal constant that appears in Newton's Law of Gravitation.
1. The acceleration due to gravity (g) is the acceleration experienced by a mass due to the Earth's gravitational field.
2. The value of G is constant throughout the universe, and its value is approximately 6.67 × 10-11 Nm2/kg2.
2. The value of g varies from place to place on the Earth's surface. On the surface of the Earth, it is approximately 9.8 m/s2.
3. G is a fundamental constant of nature.
3. g is dependent on the mass of the Earth and the distance from the Earth's center.
4. The value of G does not change with the location on Earth.
4. The value of g decreases with altitude and increases with depth under Earth's surface.
5. Gravitational constant is used to calculate the force between two point masses using the equation F = Gm1m2
/ d2.
5. Acceleration due to gravity (g) is used to calculate the weight of an object using the equation W = mg, where W is weight and m is the mass of the object.
/ d2.
(b) Mass and Weight
Difference Between Mass and Weight
Mass
Weight
1. Mass is a measure of the amount of matter in an object.
1. Weight is the force exerted by gravity on an object.
2. Mass is a scalar quantity and does not have a direction.
2. Weight is a vector quantity as it has both magnitude and direction (direction towards the center of the Earth or any other gravitational source).
3. The mass of an object is constant and does not change with location.
3. The weight of an object varies depending on the gravitational field of the body where the object is located (e.g., on the Moon, weight is one-sixth of that on Earth).
4. The SI unit of mass is kilogram (kg).
4. The SI unit of weight is Newton (N), which is a unit of force.
5. Example: A person has a mass of 70 kg everywhere, regardless of location.
5. Example: The same person weighs about 686 N on Earth but will weigh only 114 N on the Moon.
3. Give reason:
(a) Acceleration due to gravity is not the same in all parts of the earth.
(b) Jumping from a significant height may cause more injury.
(c) Mass of Jupiter is about 319 times the mass of the Earth, but its acceleration due to gravity is only about 2.6 times the acceleration due to gravity of the Earth.
(d) Among the objects dropped from the same height in the polar region and the equatorial region of the earth, the object dropped in the polar region falls faster.
(e) Out of two paper sheets, one is folded to form a ball. If the paper ball and the sheet of paper are dropped simultaneously in the air, the folded paper will fall faster.
(f) When a marble and a feather are dropped simultaneously in a vacuum, they reach the ground together (at the same time).
(g) As you climb Mount Everest, the weight of the goods that you carry decreases.
(h) It is difficult to lift a big stone on the surface of the earth. but it is easy to lift a smaller one.
(i) Mass of an object remains constant but its weight varies from place to place.
(j) One will have an eerie feeling when he/she moves down while playing a Rote Ping.
4. Answer the following questions:
(a) What is gravity?
(b) State Newton's universal law of gravitation.
(c) Write the nature of gravitational force.
(d) Define gravitational constant (G).
(e) Under what conditions is the value of gravitational force equal to the gravitational constant (F=G)?
(f) Write two effects of gravitational force.
(g) Mathematically present the difference in the gravitational force between two objects when the mass of each is made double and the distance between them is made one forth their initial distance.
and the initial masses, and the initial distance be
. The initial gravitational force is given by:
Now, the masses are doubled, so the new masses are and , and the distance is reduced to one-fourth, so the new distance is . The new gravitational force is:
Simplifying the equation:
Thus,
Therefore, the final gravitational force is 64 times the initial gravitational force.
(h) What is gravitational force?
(i) Define acceleration due to gravity.
(j) What is free fall? Give two examples of it.
(k) Under what conditions is an object said to be in free fall?
(l) Write the conclusions of the feather and coin experiment.
(m) What is weightlessness?
(n) Mention any four effects of gravitational force.
(o) Prove that acceleration due to the gravity of the Earth is inversely proportional to the square of its radius (g ∝ 1 / R)
) between two masses and is given by:
Where:
- is the force between the two masses,
- is the gravitational constant (
Let's consider an object of mass placed on or near the surface of a earth (with mass and radius ). The force of gravity acting on the object due to the planet will be the gravitational force between the planet and the object.
In this case, we have:
- (mass of the planet),
- (mass of the object),
- e distance from the center of the planet to the object).
Thus, the gravitational force acting on the object is:
Since this is the force due to gravity acting on the object.
According to Newton's Second Law of Motion:
Since both expressions represent the force on the object, we can set them equal to each other:
This is the desired formula for the acceleration due to gravity at the surface of the planet. It tells us that:
or, g ∝ 1 / R2 (If G and M are taken as constant)
This equation expresses that the acceleration due to gravity 'g' is inversely proportional to the square of the radius of the planet (R).
(p) Mention the factors that influence acceleration due to gravity.
(q) The acceleration due to the gravity in the Earth surface is 9.8 m/s2. What does this mean?
(r) Mass of the Moon is about 1/81 times the mass of the Earth and radius of earth is about 3.7 times that of moon. If the earth is squeezed to the size of the moon, what will be the effect on its acceleration due to gravity? Explain with the help of mathematical calculation.
(s) The acceleration due to gravity of an object of mass 1 kg in outer space is 2m/s². What is the acceleration due to gravity of another object of mass 10 kg at the same point? Justify with arguments.
(t) A man first measures the mass and weight of an object in the mountain and then in the Terai. Compare the data that he obtains.
(u) A student suggests a trick for gaining profit in a business. He suggests buying oranges from the mountain selling them to Terai at the cost price. If a beam balance is used during this transaction, explain, based on scientific fact, whether his trick goes wrong or right.
(v) How is it possible to have a safe landing while jumping from a flying airplane using a parachute? Is it possible to have a safe landing on the moon in the same way? Explain with reasons.
(w) The acceleration of an object moving on the earth is inversely proportional to the mass of the object, but for an object falling towards the surface of the earth, the acceleration does not depend on the mass of the object, why?
5. Solve the following mathematical problems: [Note: For best view of numericals open the page in this way]
(a) The masses of two objects A and B are 20 kg and 40 kg respectively. If the distance between their centers is 5 m, calculate the gravitational force produced between them.
→ Solution: Mass of object A = m1 = 20 kg
Mass of object B = m2 = 40 kg
Distance between their centers (d) = 5 m
Now,
Gravitational force is given by F = Gm1m2 / d2 = (6.67 × 10-11) × (20 × 40) / 5² = (6.674 × 10-11) × 800 / 25 = (6.674 × 10-11) × 32 = 2.14 × 10-9 N
(b) Calculate the gravitational force between the two bodies shown in the figure.
→ There's no any figure given in book without which calculation is not possible.
(c) Mass of the Sun and Jupiter are 2 × 1030 kg and 1.9 × 1027 kg respectively. If the distance between Sun and Jupiter is 1.8 × 108 km, calculate the gravitational force between Sun and Jupiter.
→ Solution: Mass of sun = m1 = 2 × 1030 kg
Mass of Jupiter = m2 = 1.9 × 1027 kg
Distance between their centers = d = 1.8 × 108 km = 1.8 × 1011 m Now, Gravitational force is given by F = Gm1m2 / d2 = (6.67 × 10-11) × (2 × 1030 × 1.9 × 1027) / (1.8 × 1011)²
= (6.67 × 10-11) × (3.8 × 1057) / 3.24 × 1022
F ≈ 7.82 × 1024 N
= (6.67 × 10-11) × (3.8 × 1057) / 3.24 × 1022
F ≈ 7.82 × 1024 N
(d) Gravitational force produced between the Earth and Moon is 2.01 × 1020 N If the distance between these two masses is 3.84 × 105 km and the mass of the earth is 5. 972 × 1024 kg, calculate the mass of the moon.
→ Solution:
Mass of earth = m1 = 5. 972 × 1024 kg
Mass of moon = m2 = ?
Gravitational force = F = 2.01 × 1020 N Distance between their centers (d) = 3.84 × 105 km = 3.84 × 108 m Now, Gravitational force = F = Gm1m2 / d2 or, m2 = Fd2 / Gm1 = 2.01 × 1020 (3.84 × 106)2 / (6.67 × 10-11).(5. 972 × 1024) = 7.67 × 1022 kg.
(e) Gravitational force produced between the Earth and the Sun is 3.54 × 1022 N If the masses of the Earth and sun are 5.972 × 1024 kg and 2 × 1030 kg respectively, what is the distance between them?
→ Solution:
Mass of earth = m1 = 5.972 × 1024 kg
Mass of sun = m2 = 2 × 1030 kg Gravitational force = F = 3.54 × 1022 N Distance between their centers = d = ? Now, Gravitational force = F = Gm1m2 / d2 = (6.67 × 10-11).(5.972 × 1024).(2 × 1030) / (3.54 × 1022) d2 = 2.25 × 1022 or, d = 1.5 × 1011 m
(f) The mass of the moon is 7.342 × 1022 kg. If the average distance between the earth and the moon is 384400 km, calculate the gravitational force exerted by the moon on every kilogram of water on the surface of the earth.
→ Solution: Mass of water on earth = m1 = 1 kg Mass of moon = m2 = 7.342 × 1022 Gravitational force (F) = ? Distance between their centers (d) = 384400 km = 384400000 m Now, Gravitational force = F = Gm1m2 / d2 or, F = (6.67 × 10-11) . (1) . (7.342 × 1022) / (384400000)2 = 3.314 × 10-5 N
(g) If the mass of the moon is 7.342 × 1022 kg and its radius is 1737 km. calculate its acceleration due to gravity.
→ Solution: Mass of Moon (M) = 7.342 × 1022 Radius (r) = 1737 km = 1737000 m
Acceleration due to gravity (g) = ?
We know that,
g = GM / r² or, g = (6.67 × 10-11) . (7.342 × 1022) / (1737000)² = 1.623 m/s²
(h) Mass of the Earth is 5.972 × 1024 kg and the diameter of the moon is 3474 km. If the earth is compressed to the size of the moon, how many times will be the change in acceleration due to the gravity of the earth so formed than that of the real Earth?
→ Solution: Mass of Earth (M) = 5.972 × 1024 kg [it remains unchanged] We know the diameter of earth is 12742 km. So, Original radius of Earth = Ro = 12742/2 = 6371 km = 6371000 m Diameter of Moon = 3434 km = 3474000 m
So, Radius of Moon = 3474000/2 = 1737000 m
According to question, radius of moon = Compressed radius of Earth = Rc = 1737000 m
Now, Acceleration due to gravity (g) = GM / R² Since, the mass of earth remains same before and after compression, ratio of acceleration due to gravity before and after compression is given by
gc / go = (Ro)² / (Rc)² or, gc / go = (6371000)² / (1737000)² ∴ gc / go = 13.45Hence, new acceleration due to gravity of earth will be 13.45 times the acceleration due to gravity of earth before compression.
(i) If the mass of Mars is 6.4 × 1023 kg and its radius is 3389 km, calculate its acceleration due to gravity. What is the weight of an object of mass 200 kg on the surface of Mars?
→ Solution: Mass of Mars (M) = 6.4 × 1023 kg Radius of mars (r) = 3389 km = 3389000 m
We know, Acceleration due to gravity (g) = GM/r2 = 6.67 × 10-11 × 6.4 × 1023 / (3389000)2 = 42.688 × 1012 / (3.389 × 106)2 = 42.688 / 1.1485 × 10 = 3.716 m/s2
Again, Mass = 200 kg, Weight (W) = ?
We know, Weight (W) = Mass × gravity = 200 × 3.716 = 743.2 N(j) The acceleration due to the gravity of the earth is 9.8 m/s2. If the mass of Jupiter is 319 times the mass of the Earth and its radius is 11 times the radius of the Earth, calculate the acceleration of gravity of Jupiter. What is the weight of an object of mass 100 kg on Jupiter?→ Solution:
Earth’s acceleration due
to the gravity (g) = 9.8 m/s2
Mass of Earth = m (say)
Mass of Jupiter = 319m
Radius of earth = r (say)
Radius of Jupiter = 11r
Acceleration due to
gravity at Jupiter = gj (say)
Now,
For earth: g
= Gm/r2 → 9.8 = Gm/r2 ……. (i)
For Jupiter:
gj = G × 319m /(11r)2 ………. (ii)
Dividing equation (ii) by (i), we get: gj / 9.8 =
319/121 → gj = 25.83 m/s Again, Weight of object of mass 100 kg (W) = mass × gravity = 100 × 25.83 = 2583 N
(k) Earth's mass is 5.972×1024 kg and its radius is 6371 km. Calculate the acceleration due to the gravity of the earth at the height of the artificial satellite shown in the figure.
→ There's no any figure given in book without which calculation is not possible.
(l) Mass of the earth is 5.972×1024 kg and its radius is 6371 km. If the height of Mt. Everest is 8848.86 m from the sea level, calculate the weight of an object of mass 10 kg at the peak of Mt. Everest.
(m) The acceleration due to gravity of the Mars is 3.75m/s². How much mass can a weight-lifter lift on Mars who can lift 100 kg mass on the Earth?
→ Solution: To calculate how much mass a weight-lifter can lift on Mars, we use the relationship:Where, W= weight , m= mass and g= acceleration due to gravity]
Let us first determine his weight-lifting capability on Earth
The weight-lifter can lift a mass of on Earth.
On Earth, the acceleration due to gravity is 2
The corresponding weight is:
×Now, let us relate the same weight to Mars gravity:
On Mars, the acceleration due to gravity is 2
Using the formula
The weight-lifter who can lift
(o) If a stone is dropped from a height of 15 m. how long will it take to reach the ground? Calculate the velocity of the stone when it hits the ground.
→ Solution: height from which stone is dropped (h) = 15 m
initial velocity (u) = 0 m/s
time taken to reach ground (t) = ?
velocity of stone when it hits the ground (v) = ?
Now,
For object in free fall, h = ut + 1/2 (gt2) or, 15 = 0 + 1/2 (9.8 × t2) or, 15 / 4.9 = t2 or, t = 1.749 s
Also, For final velocity of stone, we use (v) = u + gt = 0 + 9.8 × 1.749 = 17.14 m/s
Hence, the stone will take 1.749 s to reach the ground and its velocity will be 17.14 m/s when it hits the ground.
(p) If a cricket ball is thrown vertically upwards into the sky with a velocity of 15 m/s, to what maximum height will the ball reach?→ Solution: Intial velocity (u) = 15 m/s
acceleration of the ball (g) = -9.8 m/s2 maximum height to which ball reach (h) =?
Now, we know, v = u + gt or, 0 = 15 + (-9.8) t [Since, velocity at maximum height is 0, so v =0] or, t = 1.53 s
Again, we know, h = ut + 1/2 gt2 or, h = 15 × 1.53 + 1/2 (-9.8 × 1.53 × 1.53) or, h = 22.95 - 1/2 × 22.94 h = 22.95 - 11.47 = 11.48 m
Hence, The ball will reach height of 11.48 m in maximum.
QUESTIONS ASKED IN SEE
- Birth of Stars: Stars are born when gravity pulls together clouds of gas and dust in space. As these particles come closer, the temperature and pressure increase until nuclear fusion starts, forming a star. Without gravity, this process would not occur.
- Death of Stars: When a star runs out of fuel, gravity causes it to collapse. Depending on the mass of the star, it might become a white dwarf, neutron star, or black hole. This process would not be possible without the force of gravity.
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