In the previous section, we saw the forces exerted by a spherical shell on a point mass inside or outside it. In the discussions so far in this chapter, we came across the universal gravitational constant (G) many times. We will now see how scientists determined the actual value of G
In 1798, the British scientist Henry Cavendish set up an apparatus called the torsion balance. His aim was to determine the density of the Earth. Though he succeeded in finding the density, he did not find it necessary to determine the value of G. However, the same setup was used by later scientists to calculate G accurately. We will write the procedure in steps:
1. Fig.8.22 below shows the schematic arrangement of the setup
• There are four balls
♦ The two small green balls are identical. They have the same diameter
♦ The two large magenta balls are identical. They have the same diameter
• All the four balls are made of lead
2. The green balls are attached to the ends of a rod
• The rod is suspended using a thin wire. This thin wire is shown in blue color
♦ The top end of this wire is attached to a rigid support
♦ The bottom end of this wire is attached at point O
3. The following 3 conditions should be satisfied:
(i) Consider an imaginary line joining the centers of the green balls
• Let the length of this line be l
• Point O must be the exact midpoint of this imaginary line
(ii) Consider an imaginary line joining the centers of the magenta balls
• Length of this imaginary line should be the same l mentioned in (i)
• Point O must be the exact midpoint of this imaginary line also
(iii) The green and magenta balls should be placed on alternate sides. This can be explained using the fig.8.22
• On the left side, the green ball is placed at the rear of the magenta ball
• On the right side, the green ball is placed at the front of the magenta ball
4. When the conditions 3(i) and 3(ii) are satisfied, the centers of all the four spheres will lie on an imaginary circle with center at O
• This imaginary circle is indicated by the dashed yellow curve
5. The setup is complete. Now we can begin the experiment
• Let r be the initial distance between any one magenta ball and it's corresponding green ball. This is shown in fig.8.22
• Due to the gravitational force of attraction, the green balls will move towards the magenta balls
6. The magnitude of the force acting on any one green ball is given by:
$\mathbf\small{|\vec{F}|=\frac{GMm}{r^2}}$
• Where:
♦ M is the mass of the larger magenta ball
♦ m is the mass of the smaller green ball
• This same magnitude acts on both the green balls. But in opposite directions
7. Since the forces are equal in magnitude but opposite in directions, a torque is created
• Magnitude of this torque = Magnitude of any one force × Perpendicular Distance between the forces
= $\mathbf\small{|\vec{F}|l=\frac{GMml}{r^2}}$
8. The green balls will not touch the magenta balls
• But that is unexpected. The torque must rotate the two green balls and bring them into contact with with the magenta balls
• So what is stopping the green balls ?
The answer can be written in 8 steps:
(i) When the green balls rotate, the rod attached to them also rotates
(ii) When the rod rotates, the blue wire also rotates
(iii) But the blue wire is fixed firmly at it's top end
So the wire is not 'free to rotate'
(iv) The bottom portions of the wire rotate. But the top portions resist rotation
• As a result, the wire is twisted
(v) When the wire is twisted, it tries to resist 'being twisted'
• This is called torsional resistance
• In our present case, 'torsional resistance' means, the torque which is resisting the rotation of the green balls
• This torsional resistance is indicated by the white curved arrow in fig.8.23 below:
(vi) In the fig.8.23, note the two directions:
♦ Direction of rotation of the green balls
♦ Direction of the white curved arrow
• The above two directions are opposite to each other. This is indeed so because, the torsional resistance will be opposing the rotation of the green balls
(vii) We calculated the torque created by $\mathbf\small{|\vec{F}|}$ in (7)
• At the initial stages, this torque is able to over come the torsional resistance offered by the wire
• So the green balls move towards the magenta balls
(viii) But as the rotation continues, the torsional resistance increases
• It increases to such a level that, it becomes equal to the value calculated in (7)
• At that stage, the rotation stops
• So at that stage, we can write:
Torsional resistance offered by the wire = $\mathbf\small{\frac{GMml}{r^2}}$
9. Consider the equation written in 8(viii) above
• If we can find the 'torsional resistance offered by the wire' at the final stage, we will be able to calculate G
• This is because, all others are known quantities
• So our next aim is to find this 'torsional resistance'
• For that, we have to perform a separate experiment. The following steps from (10) to (14) give a basic idea about that experiment
10. In fig.8.24 below, a blue cylinder is fixed at one of it's ends. The other end is free
11. In the fig.8.25 below, a torsion is applied at the free end
• This torsion is indicated by the cyan curved arrow
• Consider any fibre on the surface, say PQ
• When torsion is applied, the end Q will move to a new position Q'
• But since the other end of the cylinder is fixed, P remains at the same position
12. So we get an angle Q'PQ
• This angle is called the angle of twist (θ)
• It is measured in radians
♦ If the cylinder is strong, we will get only a small θ even if we apply a large torsion
♦ If the cylinder is weak, we will get a large θ even if we apply a small torsion
• Thus, θ is a property of the object
13. A number of trials are done on the cylinder
• In each trial, a known torsion is applied and the corresponding θ is noted
■ From that data, we get an important information:
The exact torsion ($\mathbf\small{\tau}$) required to obtain a θ of 1 radian
■ Each object has it's own unique value of $\mathbf\small{\tau}$
14. In our present case, the 'object of interest' is the blue wire used for suspending the green balls
• So the experiment is performed on the blue wire and it's $\mathbf\small{\tau}$ is determined
15. Now we get back to our main experiment
• We have the initial and final positions of any one of the two green balls
• From those positions, we can determine the angle (θ') through which the blue wire is twisted
• Also we have $\mathbf\small{\tau}$, which is the unique property of the wire
16. So, if we multiply θ' by $\mathbf\small{\tau}$, we will get the exact torsional resistance
• That means: Torsional resistance = $\mathbf\small{\tau \theta'}$
17. So the equation in 8(v) becomes: $\mathbf\small{\tau \theta'=\frac{GMml}{r^2}}$
• In this equation, G is the only unknown. So it can be easily calculated
In 1798, the British scientist Henry Cavendish set up an apparatus called the torsion balance. His aim was to determine the density of the Earth. Though he succeeded in finding the density, he did not find it necessary to determine the value of G. However, the same setup was used by later scientists to calculate G accurately. We will write the procedure in steps:
1. Fig.8.22 below shows the schematic arrangement of the setup
 |
| Fig.8.22 |
♦ The two small green balls are identical. They have the same diameter
♦ The two large magenta balls are identical. They have the same diameter
• All the four balls are made of lead
2. The green balls are attached to the ends of a rod
• The rod is suspended using a thin wire. This thin wire is shown in blue color
♦ The top end of this wire is attached to a rigid support
♦ The bottom end of this wire is attached at point O
3. The following 3 conditions should be satisfied:
(i) Consider an imaginary line joining the centers of the green balls
• Let the length of this line be l
• Point O must be the exact midpoint of this imaginary line
(ii) Consider an imaginary line joining the centers of the magenta balls
• Length of this imaginary line should be the same l mentioned in (i)
• Point O must be the exact midpoint of this imaginary line also
(iii) The green and magenta balls should be placed on alternate sides. This can be explained using the fig.8.22
• On the left side, the green ball is placed at the rear of the magenta ball
• On the right side, the green ball is placed at the front of the magenta ball
4. When the conditions 3(i) and 3(ii) are satisfied, the centers of all the four spheres will lie on an imaginary circle with center at O
• This imaginary circle is indicated by the dashed yellow curve
5. The setup is complete. Now we can begin the experiment
• Let r be the initial distance between any one magenta ball and it's corresponding green ball. This is shown in fig.8.22
• Due to the gravitational force of attraction, the green balls will move towards the magenta balls
6. The magnitude of the force acting on any one green ball is given by:
$\mathbf\small{|\vec{F}|=\frac{GMm}{r^2}}$
• Where:
♦ M is the mass of the larger magenta ball
♦ m is the mass of the smaller green ball
• This same magnitude acts on both the green balls. But in opposite directions
7. Since the forces are equal in magnitude but opposite in directions, a torque is created
• Magnitude of this torque = Magnitude of any one force × Perpendicular Distance between the forces
= $\mathbf\small{|\vec{F}|l=\frac{GMml}{r^2}}$
8. The green balls will not touch the magenta balls
• But that is unexpected. The torque must rotate the two green balls and bring them into contact with with the magenta balls
• So what is stopping the green balls ?
The answer can be written in 8 steps:
(i) When the green balls rotate, the rod attached to them also rotates
(ii) When the rod rotates, the blue wire also rotates
(iii) But the blue wire is fixed firmly at it's top end
So the wire is not 'free to rotate'
(iv) The bottom portions of the wire rotate. But the top portions resist rotation
• As a result, the wire is twisted
(v) When the wire is twisted, it tries to resist 'being twisted'
• This is called torsional resistance
• In our present case, 'torsional resistance' means, the torque which is resisting the rotation of the green balls
• This torsional resistance is indicated by the white curved arrow in fig.8.23 below:
 |
| Fig.8.23 |
♦ Direction of rotation of the green balls
♦ Direction of the white curved arrow
• The above two directions are opposite to each other. This is indeed so because, the torsional resistance will be opposing the rotation of the green balls
(vii) We calculated the torque created by $\mathbf\small{|\vec{F}|}$ in (7)
• At the initial stages, this torque is able to over come the torsional resistance offered by the wire
• So the green balls move towards the magenta balls
(viii) But as the rotation continues, the torsional resistance increases
• It increases to such a level that, it becomes equal to the value calculated in (7)
• At that stage, the rotation stops
• So at that stage, we can write:
Torsional resistance offered by the wire = $\mathbf\small{\frac{GMml}{r^2}}$
9. Consider the equation written in 8(viii) above
• If we can find the 'torsional resistance offered by the wire' at the final stage, we will be able to calculate G
• This is because, all others are known quantities
• So our next aim is to find this 'torsional resistance'
• For that, we have to perform a separate experiment. The following steps from (10) to (14) give a basic idea about that experiment
10. In fig.8.24 below, a blue cylinder is fixed at one of it's ends. The other end is free
 |
| Fig.2.24 |
• This torsion is indicated by the cyan curved arrow
 |
| Fig.8.25 |
• When torsion is applied, the end Q will move to a new position Q'
• But since the other end of the cylinder is fixed, P remains at the same position
12. So we get an angle Q'PQ
• This angle is called the angle of twist (θ)
• It is measured in radians
♦ If the cylinder is strong, we will get only a small θ even if we apply a large torsion
♦ If the cylinder is weak, we will get a large θ even if we apply a small torsion
• Thus, θ is a property of the object
13. A number of trials are done on the cylinder
• In each trial, a known torsion is applied and the corresponding θ is noted
■ From that data, we get an important information:
The exact torsion ($\mathbf\small{\tau}$) required to obtain a θ of 1 radian
■ Each object has it's own unique value of $\mathbf\small{\tau}$
14. In our present case, the 'object of interest' is the blue wire used for suspending the green balls
• So the experiment is performed on the blue wire and it's $\mathbf\small{\tau}$ is determined
15. Now we get back to our main experiment
• We have the initial and final positions of any one of the two green balls
• From those positions, we can determine the angle (θ') through which the blue wire is twisted
• Also we have $\mathbf\small{\tau}$, which is the unique property of the wire
16. So, if we multiply θ' by $\mathbf\small{\tau}$, we will get the exact torsional resistance
• That means: Torsional resistance = $\mathbf\small{\tau \theta'}$
17. So the equation in 8(v) becomes: $\mathbf\small{\tau \theta'=\frac{GMml}{r^2}}$
• In this equation, G is the only unknown. So it can be easily calculated
• So we have seen how scientists determined the value of G
• In the next section, we will see the acceleration due to gravity of the Earth
• In the next section, we will see the acceleration due to gravity of the Earth
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