Two banked curves have the same radius. Curve A is banked at an angle of 12.2°, and curve B is banked at an angle of 21.1°. A car can travel around curve A without relying on friction at a speed of 16.5 m/s. At what speed can this car travel around curve B without relying on friction?
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Hello Jesse,
This one looks interesting. We need to come up with an equation with Radius (r) on the left and Velocity, angle and maybe gravity on the right.
Draw an FDB for the car. If there is no friction then there are only two forces, the centripetal force and the weight of the car. These two forces need to be broken into components that are either parallel or perpendicular to the bank.
By taking the sum of the forces parallel to the slope MUST be zero if there is no friction
W sin alpha = F cos alpha
W = mg and F = m * a = m * V^2/r
mg sin alpha = (mV^2 /r) * cos alpha
r = (V^2 cos alpha)/(g * sin alpha)
r = V^2 / (tan alpha * g)
Now we could solve for the radius here and then use it to solve for the velocity on the second track, but they don't ask us for radius, right? Why bother to calculate it,
Lets call the bank of the second bank (beta) and the Velocity on the second bank (V2)
So our radius curve for this bank is
r = V2^2 / (tan beta * g)
Well we know that the radii of both banks are the same, so we can set the equations equal to each other.
V^2 / (tan alpha * g) = V2^2 / (tan beta * g)
Solve for V2
V2^2 = V^2 tan beta / tan alpha
V2 = sqrt (V^2 tan beta / tan alpha)
V2 = sqrt ((16.5 m/s)^2 tan 21.1 / tan 12.2)
V2 = 22.0 m/s
Tah Dahhhhh. ................. ................... .............. (crickets)
Good Luck
This one looks interesting. We need to come up with an equation with Radius (r) on the left and Velocity, angle and maybe gravity on the right.
Draw an FDB for the car. If there is no friction then there are only two forces, the centripetal force and the weight of the car. These two forces need to be broken into components that are either parallel or perpendicular to the bank.
By taking the sum of the forces parallel to the slope MUST be zero if there is no friction
W sin alpha = F cos alpha
W = mg and F = m * a = m * V^2/r
mg sin alpha = (mV^2 /r) * cos alpha
r = (V^2 cos alpha)/(g * sin alpha)
r = V^2 / (tan alpha * g)
Now we could solve for the radius here and then use it to solve for the velocity on the second track, but they don't ask us for radius, right? Why bother to calculate it,
Lets call the bank of the second bank (beta) and the Velocity on the second bank (V2)
So our radius curve for this bank is
r = V2^2 / (tan beta * g)
Well we know that the radii of both banks are the same, so we can set the equations equal to each other.
V^2 / (tan alpha * g) = V2^2 / (tan beta * g)
Solve for V2
V2^2 = V^2 tan beta / tan alpha
V2 = sqrt (V^2 tan beta / tan alpha)
V2 = sqrt ((16.5 m/s)^2 tan 21.1 / tan 12.2)
V2 = 22.0 m/s
Tah Dahhhhh. ................. ................... .............. (crickets)
Good Luck