1. Evaluate dF/dr when r=10,000, given that F= GmM/(r^2) and that F decreases by 3N/m when r=20,000.
2. The lowest point of a 5 meter long ladder is pulled away from a wall at 0.3 meters/sec. Find the rate at which the angle between the ladder and the wall changes when the ladder's lowest point is 3 meters away from the wall.
a. F=k/r^2
f'=-2k/r
df/dr=-2k/r
at r=20k
-3=-k/20000
k=60,000 N
so when r=10,000
df/dr=-60,000N/10,000m=-6N/m
CosTheta=x/5
-sinTheta*dTheta/dt=dx/dt *1/5
but sinTheta= sqrt(5^2-x^2)/25
sinTheta=sqrt(1-(x/5)^2)
solve4 for dThteta/dt
1. To evaluate dF/dr when r=10,000, we need to find the derivative of F with respect to r.
Given that F = GmM / r^2, where G is the gravitational constant, m and M are masses, and r is the distance between the masses.
First, let's find the value of GmM. We know that F decreases by 3 N/m when r=20,000. Therefore, we can write:
F(r=20,000) = GmM / (20,000)^2
F(r=20,000) = GmM / 400,000,000
We also know that F = GmM / r^2, so we can substitute F(r=20,000) into the equation:
GmM / 400,000,000 = GmM / r^2
Now, we can solve for GmM:
GmM = (400,000,000)(GmM) / r^2
Simplifying the equation, we get:
1 = 400,000,000 / r^2
Solving for r^2, we get:
r^2 = 400,000,000
Taking the square root of both sides, we find:
r = √(400,000,000)
r = 20,000
So, GmM = 400,000,000.
Now, let's differentiate F with respect to r:
dF/dr = (d/dx)(GmM / r^2)
Using the power rule of differentiation, the derivative of r^2 with respect to r is 2r:
dF/dr = -(GmM)(2r) / r^4
At r = 10,000, plugging in the values:
dF/dr = -(400,000,000)(2(10,000)) / (10,000)^4
dF/dr = -8,000,000,000,000 / 1,000,000,000,000,000
dF/dr = -8 N/m
Therefore, when r = 10,000, the rate of change of F with respect to r is -8 N/m.
2. To find the rate at which the angle between the ladder and the wall changes, we can use trigonometry.
Let's assume that the distance between the ladder's top end and the wall is h meters.
The length of the ladder, b, is 5 meters, and we are given that the lowest point of the ladder is pulled away from the wall at a rate of 0.3 meters/sec. So, db/dt = 0.3 m/s.
We need to find dθ/dt, the rate at which the angle between the ladder and the wall changes when the lowest point of the ladder is 3 meters away from the wall. Let's call this distance x.
Using the Pythagorean theorem, we have:
x^2 + h^2 = b^2
3^2 + h^2 = 5^2
9 + h^2 = 25
h^2 = 16
h = 4
Differentiating both sides with respect to time, we have:
2x(dx/dt) + 2h(dh/dt) = 0
Substituting the given values:
2(3)(0.3) + 2(4)(dh/dt) = 0
1.8 + 8(dh/dt) = 0
8(dh/dt) = -1.8
dh/dt = -1.8/8
dh/dt = -0.225 m/s
Therefore, when the lowest point of the ladder is 3 meters away from the wall, the rate at which the angle between the ladder and the wall changes is -0.225 m/s.
To evaluate dF/dr when r=10,000, we need to differentiate the equation F= GmM/(r^2) with respect to r.
Step 1: Start with the equation F= GmM/(r^2).
Step 2: Take the derivative of both sides of the equation with respect to r. This will give us dF/dr.
Step 3: To differentiate GmM/(r^2) with respect to r, we can use the power rule for differentiation:
dF/dr = -2GmM/(r^3)
Step 4: Substitute the value of r=10,000 into the equation to find the result.
dF/dr = -2GmM/(10,000^3)
Now, let's move on to the second question.
To find the rate at which the angle between the ladder and the wall changes when the ladder's lowest point is 3 meters away from the wall, we can use trigonometry and related rates.
Step 1: Let's define the angle between the ladder and the wall as θ.
Step 2: We know that the ladder is 5 meters long and the rate at which the lowest point of the ladder is pulled away from the wall is 0.3 meters/sec.
Step 3: We want to find the rate at which the angle θ changes when the ladder's lowest point is 3 meters away from the wall. This can be represented as dθ/dt.
Step 4: We can use the tangent function to relate the angle θ with the distance between the ladder and the wall:
tan(θ) = (height of ladder) / (distance from the wall)
Step 5: Differentiate both sides of the equation with respect to time t:
sec^2(θ) * dθ/dt = (dh/dt) / (distance from the wall)
Step 6: Rearrange the equation to solve for dθ/dt:
dθ/dt = (dh/dt) * cos^2(θ) / (distance from the wall)
Step 7: Substitute the given values into the equation:
dθ/dt = (0.3 m/s) * cos^2(θ) / (3 m)
Now, you can plug in the specific value of θ for your problem and calculate the rate at which the angle between the ladder and the wall changes.