3 and 4 Marks Questions

Mass of the Jupiter is 1.9 × 10³⁰ kg and mass of the Sun is 2 × 10³⁰ kg. Calculate the distance between them if the gravitational force between them is 4.3 × 10²⁹ N.
Here,
Mass of the Jupiter (m₁) = 1.9 × 10³⁰ Kg
Mass of the Sun (m₂) = 2 × 10³⁰ kg
Gravitational force (F) = 4.3 × 10²⁹ N
Distance between Jupiter and the Sun (d) = ?
We know,
F = $\frac{Gm_{1}m_{2}}{d^2}$
or, 4.3 × 10²⁹ = $\frac{6.67 \times 10^{-11} \times 1.9 \times 10^{30} \times 2 \times 10^{30}}{d^2}$
or, d² = $\frac{25.346 \times 10^{49}}{4.3 \times 10^{29}}$
or, d² = 5.894 × 10²⁰
∴ Distance between Jupiter and the Sun (d) = 2.94 × 10¹⁰ m.
The gravitational force produced between any two objects kept 2.5×10⁴ km apart is 580N. At what distance should they be kept so that the gravitational force becomes half.
Here,
Gravitational force (W₁) = 580N
W₂ = 580/2 = 290N
distance (R₁) = 2.5 × 10⁴ km = 2.5 × 10⁷m
R₂ = ?
Now, we know,
$\frac{W_1}{W_2}$ = $\frac{(R_2)^2}{(R_1)^2}$
or, $(R_2)^2$ = $\frac{W_1 \times (R_1)^2}{W_2}$

= $\frac{580 \times (2.5 \times 10^7)^2}{290}$
= ${2 \times 6.25 \times 10^{14}}$
= ${12.5 \times 10^{14}}$

∴ R₂ = $\sqrt{12.5 \times 10^{14}}$ = 3.54 × 10⁷m.
A stone dropped freely from 72 m height of the tower and reaches the ground in 6 seconds. Calculate acceleration due to gravity of that stone. At what condition does the acceleration due to gravity becomes zero.
Here,
Height (h) = 72m
Initial velocity (u) = 0 m/s
time taken (t) = 6 s
acceleration due to gravity (g) = ?
Now, we have,
from equation of motion,
h = ut + $\frac{1}{2}$ gt²
or, 72 = 0 × 6 + $\frac{1}{2}$ × g × 6 × 6
or, g = $\frac{72 \times 2}{36}$
∴ g = 4 m/s²
When the object is brought to the center of the earth, the value of acceleration due to gravity is zero.

How much time does a stone take to reach on the surface of the earth drop from the height 20m and what will be its acceleration due to gravity after 3 seconds? (g = 10 m/s²)
Here,
height (h) = 20m
acceleration due to gravity (g) = 10 m/s²
time taken (t) = ?
Now, we have,
From equation of motion,
h = ut + $\frac{1}{2}$ gt²
or, 2h = gt² (u = 0)
or, t² = $\frac{2h}{g}$
= $\frac{2 \times 20}{10}$
∴ t = 2 second
Its acceleration due to gravity after 3 seconds is 0 m/s².
In the figure below, two identical metal balls A and B having equal masses are being dropped towards the surface of moon and earth. Analyze the given data and answer the following questions:
1. If both the metal balls are released simultaneously, which one does strike the ground faster? Show with calculation.
2. For the ball falling on the earth (B):
  From equation of motion,
  h = ut + $\frac{1}{2}$ gt²
  or, 2h = gt² (u = 0)
  or, t = $\sqrt{\frac{2h}{g}}$
  = 1.43 sec
  For the ball falling on the moon (A)
  t = $\sqrt{\frac{2 \times 10}{1.62}}$
  = 3.51 sec
  ∴ The ball B takes short time to strike the ground i.e. B strikes faster than A.
3. A person lifts 30kg on the surface of the earth. How much mass can he lift on the surface of the moon if he applies same magnitude of force?
4. According to en-question,
  weight lifted on the earth = weight lifted on the moon
  i.e. m₁ × g_e = m₂ × g_m
  or, 30 × 9.8 = m₂ × 1.62
  or, m₂ = $\frac{30 \times 9.8}{1.62}$
  ∴ m₂ = 181.48 kg
  Thus, the person can lift 181.48 kg on the surface of the moon.
The mass of the moon is 7.2×10²² kg and its radius 1.7×10⁶ m. Calculate the acceleration due to gravity of the moon and also calculate the weight of an object of mass 80 kg on the lunar surface.
Here,
Mass of the moon (M) = 7.2 × 10²² kg
Radius (R) = 1.7 × 10⁶ m
Gravitational constant (G) = 6.67 × 10^-11 Nm²/kg²
Acceleration due to gravity (g) = ?
Now, we know,
g = $\frac{GM}{R^2}$
= $\frac{6.67 \times 10^{-11} \times 7.2 \times 10^{22}}{(1.7 \times 10^6)^2}$
∴ g = 1.67 m/s²
Finally, mass of an object (m) = 80 kg
weight of an object (W) = m × g = 80 × 1.67 = 133.6 N
A meteor is falling towards the Earth. If mass and radius of the earth are 6×10²⁴ kg and 6.4×10³ km respectively. Find the height of meteor from the earth's surface where its acceleration due to gravity becomes 4 m/s². The weight of a body decreases in a coal mine, why?
Here,
mass of Earth (M) = 6 × 10²⁴ kg
Radius (R) = 6.4 × 10³ km = 6.4 × 10⁶ m
acceleration due to gravity (g) = 4m/sec²
Gravitational constant (G) = 6.67 × 10^-11Nm² /kg²
Now, g = $\frac{GM}{(R+h)^2}$
or, g = $\frac{6.67 \times 10^{-11} \times 6 \times 10^{24}}{(6.4 \times 10^6 + h)^2}$
or, (6.4 × 10⁶ + h)² = 10.005 × 10 ¹⁸
∴ h = 3.6 × 10⁶
Since coal mining takes place at depths, the value of g decreases at a height less than the surface of the earth. Therefore, the weight of the object in the coal mine is reduced.
The gravitational force produced between two bodies is 25N when they are at the distance of 4m. How much gravitational force is produced when they are kept 2m apart? Calculate.
Here,
1st case,
Gravitational Force (F₁) = 25N
Distance between two bodies(d₁) = 4m
Now, we know,
F₁ = $\frac{GM_{1}M_{2}}{d^2}$
Or, 25 × 16 = GM₁M₂ ---- (I)
Again, 2nd case,
Gravitational Force (F₂) = ?
Distance between two bodies(d₂) = 2m
Now,
F₂ = $\frac{GM_{1}M_{2}}{d^2}$
= $\frac{25 \times 16}{4}$ (from equation I)
∴ F₂ = 100N
Hence, 100N gravitational force is produced when they are kept 2m apart.
The mass and radius of the moon are 7.1 × 10²² kg and 1.7 × 10³ km respectively. Calculate the acceleration due to gravity on the moon's surface.
Here,
Mass of moon (M) = 7.1 × 10²² kg
Radius of moon (R) = 1.7 × 10³ km = 1.7 × 10⁶ m
Acceleration due to gravity (g) = ?
Now, we know,
g = $\frac{GM}{R^2}$

= $\frac{6.67 \times 10^{-11} \times 7.1 \times 10^{22}}{(1.7 \times 10^6 )^2}$
= $\frac{16.386}{10}$
= 1.63 m/s²
Hence. The acceleration due to gravity of the moon’s surface is 1.63 m/s².