ou build a loop-the-loop in your backyard. It has a diameter of 3.10 m.
a) How fast will you have to be moving at the top of the loop to avoid falling off?
b)If you'll be coasting from the time you get to the bottom of the loop, how fast will you have to be going at the bottom?
To solve both of these questions, we can use the principles of circular motion and centripetal force.
a) To avoid falling off at the top of the loop, the centripetal force must be equal to the gravitational force acting on you. At the top of the loop, the only force acting on you is your weight, which is equal to your mass (m) times the acceleration due to gravity (g).
The centripetal force is provided by the normal force (N) exerted by the loop on you. At the top of the loop, the net force acting on you can be calculated as the difference between the normal force and your weight.
Therefore, the net force can be written as:
Net Force = N - mg
Since the net force is equal to the centripetal force (Fc), we have:
N - mg = Fc
The centripetal force (Fc) acting on an object moving in a circle can be given by the equation:
Fc = (m * v^2) / r
Where:
m = mass of the object
v = velocity of the object
r = radius of the circle (half of the diameter)
In our case, the radius (r) is half of the diameter, so r = 3.10 m / 2 = 1.55 m.
At the top of the loop, the net force is equal to the centripetal force:
N - mg = (m * v^2) / r
Since we want to know how fast you have to be moving, we can rearrange the equation to solve for velocity (v):
v^2 = (r * g) + (N / m)
v = √[(r * g) + (N / m)]
Since we don't have information about the normal force, we need another equation to find its value.
The normal force can be calculated as the sum of the gravitational force and the centripetal force.
N = mg + Fc
Substituting the value of Fc from the centripetal force equation, we get:
N = mg + (m * v^2) / r
With this, we can solve for the velocity (v) at the top of the loop.
b) At the bottom of the loop, the normal force (N) is providing the centripetal force to keep you in circular motion. The net force is the difference between the normal force and your weight:
Net Force = N - mg
Again, the centripetal force (Fc) is given by:
Fc = (m * v^2) / r
Now, at the bottom of the loop, the net force is equal to the centripetal force:
N - mg = (m * v^2) / r
We can rearrange this equation to solve for velocity (v):
v^2 = r * g - (N / m)
v = √[r * g - (N / m)]
Using these equations, we can find the velocities required to avoid falling off at the top and bottom of the loop.