Calculating Black Hole Mass Limit for Gravitational Acceleration Change

[omegas]

Homework Statement

Calculate the black hole mass limit such that the change in the gravitational acceleration at a distance 50R_s over a small interval of 2m doesn't exceed 10m/s². Use Newton's Theory of Gravity in the calculation.

Homework Equations

Schwarszchild Radius:

R_s = 2GM / c²

Newton's Theory of Gravity:

F = GMm / r²

The Attempt at a Solution

Totally lost. I'm an American study abroad student in New Zealand and am completely lost with these assignments. Don't know where to go for help.

 

[BerryBoy]

Firstly, write down the expression for how the acceleration due to gravity changes with distance (from Newton's equations), then times this by 2 [metres] and force it to equal ten (the limit that the field can change) ie

$$\Delta r \left. \frac{\rm{d} g}{\rm{d} r} \right\vert_{r=50 R_\rm{s} }} =10 \rm{\, ms}^{-2}$$

where $$\Delta r = 2 \rm{\, m}$$

Does this help??

 

[omegas]

Thanks. I did the calculations, but I keep getting a negative number. I set my r = 50R_s and for my dg/dr I get -2GMr^-3. So when I solve for M is keep getting a negative number.

 

[BerryBoy]

A change in the gravitational acceleration can be positive or negative and we are not told whether to move 2 m away or towards the black hole, so solve for the absolute change (i.e. don't worry about the negative sign the solution still answers the problem).

 

[paranormal]

Hello, as a scientist, I can help you with this problem. First, let's break down the problem and define the variables given. The problem is asking for the black hole mass limit, which we will call M. The change in gravitational acceleration is given as 10m/s2, and the distance at which this change occurs is 50Rs. We are also given the value for Rs, which represents the Schwarszchild radius of the black hole.

To solve this problem, we will use Newton's Theory of Gravity, which states that the force of gravity between two objects is equal to the product of their masses (m and M) divided by the square of the distance between them (r), multiplied by the gravitational constant (G). Mathematically, this can be represented as F = GMm / r^2.

We can use this equation to calculate the change in gravitational acceleration at a distance of 50Rs. We know that the change in acceleration is equal to the difference between the acceleration at a distance of 50Rs and the acceleration at a distance of 50Rs plus 2m. So, we can set up the equation as follows:

Change in acceleration = GMm / (50Rs + 2m)^2 - GMm / (50Rs)^2

Next, we can plug in the values for the gravitational constant (G), the mass of the smaller object (m), and the distance (50Rs and 50Rs + 2m) into the equation. This will give us the change in acceleration in terms of M, the mass of the black hole.

Change in acceleration = G * m * M / (50Rs + 2m)^2 - G * m * M / (50Rs)^2

Now, we can set this change in acceleration equal to 10m/s2, as given in the problem. This will allow us to solve for M, the mass of the black hole.

10m/s2 = G * m * M / (50Rs + 2m)^2 - G * m * M / (50Rs)^2

Solving for M, we get:

M = (10m/s2 * (50Rs + 2m)^2) / (G * m) + (G * m * M) / (50Rs)^2

Finally, we can plug in the value for Rs, which is given as 2GM /