A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to six times her weight as she goes through the dip. If r = 20.0 m, how fast is the roller coaster traveling at the bottom of the dip?
N-mg=mv^2/r
N=6mg(given)
6mg-mg=mv^2/r
5mg=mv^2/r
Cancel out the mass on both sides since it is common
5g=v^2/r
Substituting the value of g and r
G= 10m/s^2 r =20 m
V= 31.6m/s
why does G equal 10. INn other words what do you do to get G
Why did the roller coaster go to the therapist? Because it was feeling a bit "up and down"! Now, let's tackle your question with a bit of a circus twist!
To find the speed of the roller coaster at the bottom of the dip, we have to consider two forces: the gravitational force (which is responsible for the passenger's weight) and the normal force (which is the force exerted by the seat pushing upward on the passenger). At the bottom of the dip, these forces together need to be six times the passenger's weight.
First, let's find the gravitational force acting on the passenger. We know that weight = mass × gravity. Assuming Earth's gravity, g, is approximately 9.8 m/s², we can say that the weight is equal to 6 times the weight of the passenger.
Next, let's relate these forces to the roller coaster's speed at the bottom of the dip. At the bottom, the normal force is equal to the gravitational force plus an additional force due to the acceleration towards the center of the circular path. This additional force is given by the formula mass × (velocity² / radius), where velocity is the speed of the roller coaster at the bottom of the dip.
Now, we can equate the forces and solve for velocity:
Weight + Weight = Weight + (mass × (velocity² / radius))
Simplifying the equation, we get:
2 × Weight = (velocity² × mass / radius)
Let's call the passenger's weight W. We know weight = 6W, so we can substitute:
2 × 6W = (velocity² × mass / radius)
12W = (velocity² × mass / radius)
Rearranging the equation, we have:
velocity² = (12W × radius) / mass
Finally, we can solve for velocity:
velocity = √((12W × radius) / mass)
Plugging in the given values: radius = 20.0 m, and the mass of the passenger is not given, we unfortunately can't provide a numerical answer. But hey, at least the passenger gets a thrilling ride!
To determine how fast the roller coaster is traveling at the bottom of the dip, we can use the concept of circular motion and centripetal force.
The centripetal force in this case is the force exerted by the seat of the car pushing upward on the passenger. We are given that this force is six times the passenger's weight. So, we can express the centripetal force as:
Centripetal force = 6 * Weight
Now, the weight of an object is given by the formula:
Weight = mass * gravity
Where gravity is the acceleration due to gravity, which is approximately 9.8 m/s^2 on Earth. Since we are not given the mass of the passenger, we can cancel it out by dividing both sides of the equation by mass, giving us:
Centripetal force (F) = 6 * Weight = 6 * mass * gravity
Next, we can relate the centripetal force to the speed and radius of the circular motion using the formula:
Centripetal force (F) = mass * (velocity^2 / radius)
By substituting the value of the centripetal force (6 * mass * gravity) into this equation, we get:
6 * mass * gravity = mass * (velocity^2 / radius)
We can now cancel out the mass on both sides of the equation:
6 * gravity = velocity^2 / radius
Finally, to isolate the velocity, we can rearrange the equation:
velocity^2 = 6 * gravity * radius
Taking the square root of both sides gives us the final equation:
velocity = √(6 * gravity * radius)
Substituting the given values:
velocity = √(6 * 9.8 m/s^2 * 20.0 m)
Simplifying the equation, we find:
velocity = √(1176) ≈ 34.3 m/s
Therefore, the roller coaster is traveling at approximately 34.3 m/s at the bottom of the dip.