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 2.6 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?
M (g + V^2/R) = 2.6 M g
V^2/R = 1.6 g
They have given you the value of R, and you know what g is.
Solve for V.
313.6
To find the speed of the roller coaster at the bottom of the dip, we can start by considering the forces acting on the passenger at that point.
1. Weight (mg): The passenger experiences a downward force due to gravity, which is equal to her mass (m) multiplied by the acceleration due to gravity (g ≈ 9.8 m/s²). So, the weight of the passenger is given by W = mg.
2. Normal force (N): The seat of the car exerts an upward force on the passenger, known as the normal force (N). In this case, the normal force is given as 2.6 times the weight of the passenger, so N = 2.6W.
3. Centripetal force (Fc): At the bottom of the dip, the net force acting on the passenger is directed towards the center of the circular path and is responsible for the circular motion. This force is called the centripetal force (Fc). In this case, the centripetal force is equal to N - mg.
Now, let's calculate the speed of the roller coaster at the bottom of the dip using these forces.
The centripetal force is given by the equation:
Fc = (mv²) / r
where m is the mass of the passenger and v is the speed of the roller coaster.
Substituting N - mg for Fc, we have:
N - mg = (mv²) / r
Since N = 2.6W and W = mg, we can write:
2.6W - mg = (mv²) / r
Rearranging the equation, we get:
2.6mg - mg = (mv²) / r
1.6mg = (mv²) / r
Canceling out the mass (m) from both sides:
1.6g = v² / r
Now, we can solve for the speed (v) at the bottom of the dip:
v² = 1.6g * r
v = √(1.6g * r)
Plugging in the values of g (acceleration due to gravity ≈ 9.8 m/s²) and r (radius = 20.0 m), we can calculate the speed (v).
v = √(1.6 * 9.8 * 20.0)
v ≈ √313.6
v ≈ 17.7 m/s
Therefore, the roller coaster is traveling at approximately 17.7 m/s at the bottom of the dip.