NCERT Class 9 Science Solutions: Gravitation
Question:
What is the importance of the universal law of gravitation?
Concept in a Minute:
The universal law of gravitation describes the force of attraction between any two objects with mass. It’s crucial for understanding celestial motion, Earth’s gravity, and tides.
Explanation:
The importance of the universal law of gravitation lies in its ability to explain and predict a wide range of phenomena:
1. Explains Celestial Motion: It provides the fundamental basis for understanding why planets orbit the sun, why moons orbit planets, and the motion of stars and galaxies. Without this law, we wouldn’t have accurate models for predicting eclipses or the paths of celestial bodies.
2. Defines Earth’s Gravity: It explains why objects fall towards the Earth. The Earth’s mass exerts a gravitational pull on everything on its surface and in its atmosphere, which we experience as weight.
3. Underpins Tides: The gravitational pull of the Moon and, to a lesser extent, the Sun on Earth’s oceans is responsible for the phenomenon of tides.
4. Foundation of Astronomy and Astrophysics: It’s a cornerstone of modern astronomy and astrophysics, forming the basis for studying the structure and evolution of the universe.
5. Predicts Motion of Satellites: It allows us to calculate the orbits of artificial satellites, crucial for communication, navigation (GPS), and space exploration.
6. Universal Applicability: The “universal” aspect means this law applies to all objects in the universe, from subatomic particles to massive stars, establishing a consistent physical principle across cosmic scales.
Question:
What happens to the force between two objects, if the distance between the objects is doubled and tripled?
Concept in a Minute:
The question is about the inverse square law of gravitation or electrostatic force. The force between two objects is inversely proportional to the square of the distance between them. This means if the distance increases, the force decreases, and if the distance decreases, the force increases. The mathematical representation is F ∝ 1/r², where F is the force and r is the distance between the objects.
Explanation:
Let the initial force between two objects be F1 and the initial distance between them be r1. According to Newton’s law of universal gravitation (or Coulomb’s law for electrostatic force), the force is inversely proportional to the square of the distance between the objects. So, F1 ∝ 1/r1².
Case 1: The distance between the objects is doubled.
The new distance, r2, is 2*r1.
The new force, F2, will be proportional to 1/r2².
So, F2 ∝ 1/(2*r1)²
F2 ∝ 1/(4*r1²)
Since F1 ∝ 1/r1², we can write F2 ∝ (1/4) * (1/r1²)
Therefore, F2 = (1/4) * F1.
This means if the distance between the objects is doubled, the force between them becomes one-fourth of the original force.
Case 2: The distance between the objects is tripled.
The new distance, r3, is 3*r1.
The new force, F3, will be proportional to 1/r3².
So, F3 ∝ 1/(3*r1)²
F3 ∝ 1/(9*r1²)
Since F1 ∝ 1/r1², we can write F3 ∝ (1/9) * (1/r1²)
Therefore, F3 = (1/9) * F1.
This means if the distance between the objects is tripled, the force between them becomes one-ninth of the original force.
Question:
What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface? (Mass of the earth is 6 × 1024 kg and radius of the earth is 6.4 × 106 m).
Concept in a Minute:
This question requires the application of Newton’s Law of Universal Gravitation to calculate the force of attraction between two objects. The formula for gravitational force is F = G * (m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Explanation:
To find the magnitude of the gravitational force between the Earth and a 1 kg object on its surface, we use Newton’s Law of Universal Gravitation.
The formula is:
F = G * (M_earth * m_object) / R_earth^2
Where:
G = Gravitational constant = 6.674 × 10⁻¹¹ N m²/kg²
M_earth = Mass of the Earth = 6 × 10²⁴ kg
m_object = Mass of the object = 1 kg
R_earth = Radius of the Earth = 6.4 × 10⁶ m
Substitute the given values into the formula:
F = (6.674 × 10⁻¹¹ N m²/kg²) * (6 × 10²⁴ kg * 1 kg) / (6.4 × 10⁶ m)²
First, calculate the product of the masses and G:
Numerator = 6.674 × 10⁻¹¹ * 6 × 10²⁴ = 40.044 × 10¹³ N m²
Next, calculate the square of the Earth’s radius:
Denominator = (6.4 × 10⁶ m)² = 40.96 × 10¹² m²
Now, divide the numerator by the denominator:
F = (40.044 × 10¹³ N m²) / (40.96 × 10¹² m²)
F = (40.044 / 40.96) × (10¹³ / 10¹²) N
F ≈ 0.9776 × 10¹ N
F ≈ 9.776 N
Rounding to two significant figures, similar to the input data:
F ≈ 9.8 N
Therefore, the magnitude of the gravitational force between the Earth and a 1 kg object on its surface is approximately 9.8 Newtons.
Question:
Why does an object float or sink when placed on the surface of water?
Concept in a Minute:
Buoyancy, Archimedes’ Principle, Density
Explanation:
An object floats or sinks based on the comparison between its density and the density of the fluid it is placed in, specifically water in this case.
Archimedes’ Principle states that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
When an object is placed on the surface of water, two forces are acting on it:
1. Its weight (acting downwards).
2. The buoyant force (acting upwards).
If the weight of the object is greater than the buoyant force exerted by the water, the object will sink. This happens when the object’s average density is greater than the density of water. The object displaces a volume of water whose weight is less than the object’s weight.
If the weight of the object is less than or equal to the buoyant force exerted by the water, the object will float. This happens when the object’s average density is less than or equal to the density of water. The object will sink just enough to displace a volume of water whose weight is equal to the object’s weight, and the buoyant force will then balance the object’s weight.
In summary:
* Object sinks if: Weight of object > Buoyant force <=> Density of object > Density of water.
* Object floats if: Weight of object <= Buoyant force <=> Density of object <= Density of water.
Question:
Write the formula to find the magnitude of the gravitational force between the earth and an object on the surface of the earth.
Concept in a Minute:
The question asks for the formula for gravitational force. This involves Newton’s Law of Universal Gravitation, which describes the attractive force between any two objects with mass. The key components are the masses of the two objects and the distance between their centers, along with the gravitational constant.
Explanation:
To find the magnitude of the gravitational force between the Earth and an object on its surface, we use Newton’s Law of Universal Gravitation. This law states that the force of attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The formula is:
F = G * (m1 * m2) / r^2
Where:
F is the magnitude of the gravitational force.
G is the universal gravitational constant, approximately 6.674 × 10^-11 N m²/kg².
m1 is the mass of the first object (in this case, the mass of the Earth).
m2 is the mass of the second object (the object on the surface of the Earth).
r is the distance between the centers of the two objects (in this case, the radius of the Earth, as the object is on its surface).
So, for an object on the surface of the Earth, the formula becomes:
F = G * (M_earth * m_object) / R_earth^2
Where:
M_earth is the mass of the Earth.
m_object is the mass of the object on the surface.
R_earth is the radius of the Earth.
Question:
If the moon attracts the earth, why does the earth not move towards the moon?
Concept in a Minute:
Newton’s Law of Universal Gravitation states that every particle of matter in the universe attracts every other particle with a force. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. According to Newton’s Third Law of Motion, for every action, there is an equal and opposite reaction.
Explanation:
The moon does attract the earth, and simultaneously, the earth also attracts the moon with an equal and opposite force. This is a direct consequence of Newton’s Law of Universal Gravitation and Newton’s Third Law of Motion. The reason the earth does not visibly move towards the moon is due to the significant difference in their masses. While the forces are equal in magnitude, the acceleration experienced by an object is inversely proportional to its mass (as per Newton’s Second Law of Motion: F = ma, so a = F/m). Since the earth has a much larger mass than the moon, it experiences a very small acceleration towards the moon, which is imperceptible compared to the moon’s acceleration towards the earth. The moon, being less massive, accelerates much more significantly towards the earth, which is why it orbits the earth.
Question:
How does the force of gravitation between two objects change when the distance between them is reduced to half?
Concept in a Minute:
The question requires understanding Newton’s Law of Universal Gravitation. This law states that the force of gravitation between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Explanation:
Let the initial distance between the two objects be r. According to Newton’s Law of Universal Gravitation, the initial force of gravitation (F1) between them is given by:
F1 = G * (m1 * m2) / r^2
Now, the distance between the objects is reduced to half. So, the new distance (r2) is r/2.
The new force of gravitation (F2) between them will be:
F2 = G * (m1 * m2) / (r/2)^2
Let’s simplify the expression for F2:
F2 = G * (m1 * m2) / (r^2 / 4)
F2 = G * (m1 * m2) * 4 / r^2
F2 = 4 * [G * (m1 * m2) / r^2]
Comparing this with the expression for F1, we can see that:
F2 = 4 * F1
Therefore, when the distance between two objects is reduced to half, the force of gravitation between them becomes four times the original force.
Question:
Write the answer of the question with reference to laws of gravitation.
State the universal law of gravitation.
Concept in a Minute:
Newton’s Law of Universal Gravitation: This law describes the attractive force between any two objects with mass. It states that this force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Explanation:
The universal law of gravitation, as stated by Sir Isaac Newton, describes the fundamental force of attraction that exists between any two objects in the universe that possess mass. It can be mathematically expressed as:
F = G * (m1 * m2) / r^2
Where:
F represents the gravitational force between the two objects.
G is the universal gravitational constant, a fundamental constant of nature, approximately equal to 6.674 × 10^-11 N m²/kg².
m1 and m2 are the masses of the two objects.
r is the distance between the centers of the two objects.
In simpler terms, this law means that:
1. The more massive the objects are, the stronger the gravitational pull between them.
2. The farther apart the objects are, the weaker the gravitational pull between them. The force decreases rapidly with increasing distance, specifically with the square of the distance.
Question:
What happens to the force between two objects, if the mass of one object is doubled?
Concept in a Minute:
The question is about the force between two objects. This implies gravitational force as per Newton’s law of universal gravitation. The key concept is Newton’s Law of Universal Gravitation, which states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Explanation:
Let the initial force between two objects be $F$. According to Newton’s Law of Universal Gravitation, the force is given by:
$F = G * (m1 * m2) / r^2$
where $G$ is the gravitational constant, $m1$ and $m2$ are the masses of the two objects, and $r$ is the distance between their centers.
Now, if the mass of one object, say $m1$, is doubled, the new mass becomes $2*m1$. Let the new force be $F’$. The formula for the new force will be:
$F’ = G * ((2*m1) * m2) / r^2$
We can rewrite this as:
$F’ = 2 * (G * (m1 * m2) / r^2)$
Notice that the term inside the parenthesis is the original force, $F$.
So, $F’ = 2 * F$.
Therefore, if the mass of one object is doubled, the force between the two objects also doubles.
Question:
Amit buys few grams of gold at the poles as per the instruction of one of his friends. He hands over the same when he meets him at the equator. Will the friend agree with the weight of gold bought? If not, why? [Hint: The value of g is greater at the poles than at the equator].
Concept in a Minute:
The concept involved is the variation of acceleration due to gravity ($g$) with latitude and the definition of weight and mass. Weight is the force of gravity acting on an object, given by $W = mg$. Mass is a measure of the amount of matter in an object and is constant.
Explanation:
Amit buys gold at the poles and brings it to the equator. The hint states that the value of $g$ is greater at the poles than at the equator.
Weight is calculated as $W = mg$, where $m$ is the mass of the gold and $g$ is the acceleration due to gravity.
When Amit buys the gold at the poles, the weight measured is $W_{poles} = m \times g_{poles}$.
When he brings the same gold to the equator, the weight measured will be $W_{equator} = m \times g_{equator}$.
Since $g_{poles} > g_{equator}$, it implies that $W_{poles} > W_{equator}$.
Therefore, the friend at the equator will not agree with the weight of the gold bought at the poles, as the measured weight will be less at the equator than at the poles for the same mass of gold. This difference in weight is due to the variation in the acceleration due to gravity.
Question:
A stone is released from the top of a tower of height 19.6 m. Calculate its final velocity just before touching the ground.
Concept in a Minute:
This problem involves uniformly accelerated motion under gravity. The key concepts are the equations of motion, specifically the one that relates initial velocity, final velocity, acceleration, and displacement. Since the stone is released, its initial velocity is zero. The acceleration is due to gravity, which is constant.
Explanation:
We can use the following equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity (what we need to calculate)
u = initial velocity
a = acceleration
s = displacement
Given:
Height of the tower (s) = 19.6 m
The stone is released, so its initial velocity (u) = 0 m/s.
The acceleration due to gravity (a) is approximately 9.8 m/s^2 (downwards, which is in the direction of motion in this case, so we take it as positive).
Substitute the given values into the equation:
v^2 = (0 m/s)^2 + 2 * (9.8 m/s^2) * (19.6 m)
v^2 = 0 + 384.16 m^2/s^2
v^2 = 384.16 m^2/s^2
To find the final velocity (v), we take the square root of both sides:
v = sqrt(384.16 m^2/s^2)
v = 19.6 m/s
Therefore, the final velocity of the stone just before touching the ground is 19.6 m/s.
Question:
What happens to the force between two objects, if the masses of both objects are doubled?
Concept in a Minute:
Newton’s Law of Universal Gravitation states that the force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is F = G * (m1 * m2) / r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
Explanation:
Let the initial force between the two objects be F. According to Newton’s Law of Universal Gravitation, F is directly proportional to the product of their masses (m1 and m2).
So, F ∝ m1 * m2.
Now, if the masses of both objects are doubled, the new masses will be 2*m1 and 2*m2.
Let the new force be F’.
Then, F’ ∝ (2*m1) * (2*m2).
F’ ∝ 4 * m1 * m2.
Since the original force F ∝ m1 * m2, we can see that F’ ∝ 4 * (m1 * m2).
This means that the new force F’ is 4 times the original force F.
Therefore, if the masses of both objects are doubled, the force between them becomes four times the original force.
Question:
Why does a block of plastic released under water come up to the surface of water?
Concept in a Minute:
Buoyancy, Density, Archimedes’ Principle. Buoyancy is an upward force exerted by a fluid that opposes the weight of an immersed object. Archimedes’ Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. Density is mass per unit volume.
Explanation:
When a block of plastic is released under water, it experiences two main forces: its weight pulling it downwards due to gravity, and the buoyant force pushing it upwards, exerted by the water. The buoyant force is equal to the weight of the water displaced by the plastic block (Archimedes’ Principle). Plastic is generally less dense than water. This means that for the same volume, the plastic block has less mass (and therefore less weight) than the water it displaces. Consequently, the upward buoyant force acting on the plastic block is greater than its downward weight. This net upward force causes the plastic block to rise to the surface of the water.
Question:
Why is it difficult to hold a school bag having a strap made of a thin and strong string?
Concept in a Minute:
Pressure is defined as force per unit area. Pressure = Force / Area. A larger area distributes the force over a wider region, resulting in lower pressure. A smaller area concentrates the force on a tiny spot, leading to higher pressure.
Explanation:
A school bag has weight, which is a force due to gravity. When this force is exerted on the shoulder, it creates pressure. A strap made of a thin string has a very small surface area of contact with the shoulder. According to the formula for pressure (Pressure = Force / Area), if the area is very small, the pressure exerted on the shoulder will be very high, even if the force (weight of the bag) is not extremely large. This high pressure on a small area causes discomfort and pain, making it difficult to hold the bag. Conversely, a wider strap distributes the same force over a larger area, significantly reducing the pressure on the shoulder and making the bag easier to carry.
Question:
What do you mean by buoyancy?
Concept in a Minute:
Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object. It is related to the pressure difference within the fluid. Archimedes’ principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Explanation:
Buoyancy is the tendency of an object to float in a fluid (like water or air). When an object is placed in a fluid, the pressure exerted by the fluid on the bottom surface of the object is greater than the pressure exerted on the top surface. This pressure difference results in an upward force, known as the buoyant force. If this buoyant force is greater than or equal to the weight of the object, the object will float. Conversely, if the weight of the object is greater than the buoyant force, the object will sink. In simpler terms, buoyancy is the “push” from the fluid that helps lift an object.
Question:
The volume of 50 g of a substance is 20 cm3. If the density of water is 1 g cm−3, will the substance float or sink?
Concept in a Minute:
The concept needed is the relationship between density, mass, and volume, and how an object’s density compares to the density of the fluid it is placed in to determine if it will float or sink. Density is calculated as mass divided by volume (Density = Mass / Volume). An object floats if its density is less than the density of the fluid, and sinks if its density is greater than the density of the fluid.
Explanation:
First, calculate the density of the substance.
Density of substance = Mass of substance / Volume of substance
Density of substance = 50 g / 20 cm³
Density of substance = 2.5 g cm⁻³
Next, compare the density of the substance with the density of water.
Density of substance = 2.5 g cm⁻³
Density of water = 1 g cm⁻³
Since the density of the substance (2.5 g cm⁻³) is greater than the density of water (1 g cm⁻³), the substance will sink in water.
Question:
You find your mass to be 42 kg on a weighing machine. Is your mass more or less than 42 kg?
Concept in a Minute:
Mass is a fundamental property of matter and remains constant regardless of location or the instrument used to measure it. A weighing machine measures weight, which is the force of gravity acting on mass. However, weighing machines are calibrated to display mass by dividing the measured weight by the local acceleration due to gravity. Therefore, a weighing machine reading directly indicates the mass.
Explanation:
A weighing machine measures the force exerted on it, which is your weight (Weight = mass × acceleration due to gravity). However, the machine is designed to display your mass by dividing this weight by the local acceleration due to gravity. Therefore, if the weighing machine shows your mass to be 42 kg, it means your mass is indeed 42 kg. The reading on the weighing machine directly represents your mass.
Question:
Why will a sheet of paper fall slower than one that is crumpled into a ball?
Concept in a Minute:
Air Resistance (Drag Force): The force exerted by air that opposes the motion of an object through it. This force depends on factors like the object’s shape, size, and speed. Gravity: The force that pulls objects towards the Earth.
Explanation:
A sheet of paper falls slower than a crumpled ball of paper because of air resistance. The flat sheet of paper has a larger surface area exposed to the air. This larger surface area interacts with more air molecules as it falls, resulting in a greater upward force of air resistance acting against the downward force of gravity. The crumpled ball of paper, on the other hand, has a significantly reduced surface area. With a smaller surface area, it encounters less air resistance, allowing the force of gravity to pull it down more effectively, thus it falls faster.
Question:
What do you mean by free fall?
Concept in a Minute:
The key concept is gravity. Free fall is a state where an object is only under the influence of gravity. This means all other forces like air resistance are negligible or absent.
Explanation:
Free fall is defined as the motion of an object where gravity is the only force acting upon it. In reality, when an object falls through the atmosphere, it experiences air resistance, which is a force opposing its motion. However, for many practical situations, especially when dealing with dense objects falling short distances, air resistance is so small that it can be ignored. In such cases, the object is considered to be in free fall, and its acceleration is solely due to the acceleration due to gravity (g), which is approximately 9.8 m/s² on Earth. This acceleration is independent of the mass of the object.
Question:
What do you mean by acceleration due to gravity?
Concept in a Minute:
Gravity: The force of attraction between any two objects with mass.
Free Fall: The motion of an object solely under the influence of gravity.
Acceleration: The rate of change of velocity.
Explanation:
Acceleration due to gravity is the constant acceleration experienced by an object when it is in free fall, meaning it is only under the influence of Earth’s gravitational pull. Near the surface of the Earth, this acceleration is approximately 9.8 meters per second squared (m/s²) and is directed downwards, towards the center of the Earth. It means that for every second an object is falling, its downward velocity increases by about 9.8 m/s. This acceleration is independent of the mass of the object (in the absence of air resistance).
Question:
In what direction does the buoyant force on an object immersed in a liquid act?
Concept in a Minute:
Archimedes’ Principle states that when a body is immersed in a fluid (liquid or gas), it experiences an upward buoyant force equal to the weight of the fluid displaced by the body. The buoyant force is a result of the pressure difference exerted by the fluid on the submerged object. Pressure increases with depth.
Explanation:
The buoyant force on an object immersed in a liquid acts in the upward direction. This is because the pressure exerted by the liquid on the bottom surface of the object is greater than the pressure exerted on the top surface. This pressure difference creates a net upward force, which we call the buoyant force. This force opposes the weight of the object, which acts downwards.
Question:
What do we call the gravitational force between the Earth and an object?
Concept in a Minute:
Gravitational Force: The force of attraction between any two objects with mass.
Earth’s Gravitational Force: The specific gravitational force exerted by the Earth on any object near its surface.
Weight: The force exerted on an object due to gravity, which is essentially the gravitational force between the object and the Earth.
Explanation:
The gravitational force that exists between the Earth and any object is what we commonly refer to as weight. While gravity is a universal force of attraction between all objects with mass, when we talk about the specific interaction between the Earth and an object, that force is experienced and measured as the object’s weight.
Answer: Weight
Question:
What is the acceleration of free fall?
Concept in a Minute:
Free fall is the motion of an object solely under the influence of gravity. Acceleration due to gravity is the constant rate at which the velocity of an object in free fall changes.
Explanation:
The acceleration of free fall is the constant acceleration experienced by an object when it is allowed to fall under the sole influence of gravity, neglecting air resistance. This acceleration is denoted by ‘g’ and its value is approximately 9.8 m/s² on the surface of the Earth. This means that for every second an object is in free fall, its downward velocity increases by 9.8 meters per second. The acceleration due to gravity is independent of the mass of the object.
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