Question 1)
Consider two point-like objects, A and B, the first of which has four times the mass of the other (i.e. MA = 4MB). They are placed a certain distance d apart.
(i) Sketch (in 2-dimensions) the gravitational field lines that are present in the region of
space around the two masses. Show the direction of the field on the field lines.
(ii) Explain quantitatively where you should place a test particle such that it experiences a net gravitational force of zero from the two point-like objects. Indicate this position on your diagram.
d is distance apart
call x distance from 4 M
then (d-x) is distance from M
4M/x^2 = M/(d-x)^2
4(d-x)^2 = x^2
4(d^2 -2dx + x^2 ) = x^2
4 d^2 -8dx +3 x^2 = 0
(2d-3x)(2d-x)=0
x = 2 d (beyond M away from 4M)
x = (2/3) d (between them but closer to M)
In the answer, can you explain how you started at 4M/x^2 = M/(d-x)^2?
Thanks
Oh, gravity, what a mysterious force. Now, let's have some fun tackling this question!
(i) Alright, let me sketch some field lines for you. Imagine both point-like objects as little stars, A and B. Since A has four times the mass of B, we'll make A bigger than B, for comedic effect, of course. Now, we'll draw the field lines for the gravitational force.
We start by drawing an arrow pointing away from A, since it has the larger mass. The arrow gets smaller as we move farther away, just like my enthusiasm for doing math. Then, we draw an arrow pointing towards B, because, poor little B, it's got less mass. The arrow pointing towards B also gets smaller as we move away from it.
These field lines end up looking like a circus tent, with A as the main attraction and B as the sad, unnoticed clown. Poor B! Just like in a circus, the field lines point towards the more massive A, emphasizing its gravitational dominance.
(ii) Ah, the sweet spot where gravity cancels out. Imagine placing a test particle somewhere between A and B, maybe right in the middle. In this position, the gravitational forces from both A and B pull towards each other, but with equal magnitude.
Think of it as a teeter-totter situation. If the forces exerted by A and B are exactly balanced, our test particle won't experience any net gravitational force. It will just sit comfortably, enjoying the magnificent circus show without any worries.
On our lovely diagram, this sweet spot can be denoted by a little smiley face, representing the test particle. It's like a "no gravity" oasis amidst the weird and wild gravitational forces around A and B.
I hope my humorous approach helped make this gravitational question a little more entertaining!
(i) To sketch the gravitational field lines in the region of space around the two masses, we can start by drawing the two point-like objects, A and B. Let's say object A is the one with four times the mass of object B.
The field lines always point away from the object. Since object A has more mass, its gravitational field lines would be longer and more spread out compared to the gravitational field lines of object B.
So, we can draw a few representative field lines coming out from object A and going towards object B. These lines should be curved, indicating that the force is directed towards object A. Similarly, we can draw a few representative field lines coming out from object B and going towards object A. These lines should also be curved, indicating that the force is directed towards object B.
(ii) To find a position where a test particle experiences a net gravitational force of zero from the two point-like objects, we need to consider the superposition of the gravitational forces from both objects.
Since both objects have point-like masses, we can use the concept of gravitational field strength (also called acceleration due to gravity) where the gravitational field strength is inversely proportional to the square of the distance from the object.
If the test particle is placed at a distance, let's say x, from object A, then it would experience a gravitational force towards object A. The magnitude of this force can be calculated using Newton's law of gravitation:
F_A = (G * MA * m) / x^2
Similarly, the test particle would also experience a gravitational force towards object B. The magnitude of this force can be calculated using the same formula:
F_B = (G * MB * m) / (d - x)^2
To find the position where the net gravitational force is zero, we need to equate these two forces:
F_A = F_B
(G * MA * m) / x^2 = (G * MB * m) / (d - x)^2
Simplifying this equation, we get:
MA / x^2 = MB / (d - x)^2
Since MA = 4MB, we can substitute this value:
4MB / x^2 = MB / (d - x)^2
Multiply both sides by (d - x)^2 to eliminate the denominators:
4MB * (d - x)^2 = MB * x^2
Expanding and simplifying:
4MB * (d^2 - 2dx + x^2) = MB * x^2
Cancel out the MB terms:
4(d^2 - 2dx + x^2) = x^2
Expand:
4d^2 - 8dx + 4x^2 = x^2
Rearranging terms:
3x^2 + 8dx - 4d^2 = 0
This is a quadratic equation in terms of x, the distance from object A. We can solve this equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Where a = 3, b = 8d, and c = -4d^2.
Solving this equation will give us two values for x, positive and negative. The positive value of x represents the position where the net gravitational force is zero. Take this positive value of x and place the test particle at that distance from object A. This position can be indicated on the sketch from part (i).
To answer this question, we need to understand the concept of gravitational fields and how they interact with test particles. Here's how we can tackle each part of the question:
(i) Sketching gravitational field lines:
Gravitational field lines represent the direction and strength of the gravitational field in a given region of space. The field lines always point towards the object that is generating the gravitational field. It means the field lines will be radially outward from the objects of masses A and B.
For this particular scenario, we have two point-like objects A and B, with A having four times the mass of B. Let's assume that object A is located to the left, and object B is located to the right.
To sketch the field lines, we start by drawing arrows pointing radially outward from both A and B. These arrows should be longer for object A than for object B, due to the difference in their masses.
Since object A has four times the mass of object B, the field lines at any given distance from A will be four times stronger compared to those from B. Therefore, the arrows stemming from object A should be four times longer than the arrows stemming from B.
The spacing between the field lines can be roughly equidistant to indicate a uniform strength of the gravitational field. Make sure to draw the field lines in a symmetric pattern, considering the direction and relative strength of each field line from A and B.
(ii) Finding the position of the test particle with zero net gravitational force:
To find the position where a test particle experiences a net gravitational force of zero, we need to consider the gravitational forces exerted by objects A and B on the test particle.
Given that the mass of A is four times the mass of B, the gravitational force exerted by A on the test particle will be four times stronger than the force exerted by B at any given distance.
At a certain location, the force due to A and the force due to B will balance each other out, resulting in a net gravitational force of zero.
The position where this happens is called the "Lagrangian point". In this scenario, the Lagrangian point lies on the line connecting A and B and is closer to object B. You can indicate this point on your diagram by placing a dot on the line, closer to object B.
To summarize, to sketch the gravitational field lines, draw arrows radially outward from objects A and B, with longer arrows for object A due to its greater mass. Draw equidistant field lines symmetrically. To find the position with zero net gravitational force, locate the Lagrangian point on the line connecting A and B, closer to object B.