A circular automobile racetrack is banked at an angle theta such that no friction between road and tires is required when a car travels at 28.0 m/s. If the radius of the track is 390 m, determine theta .
http://www.batesville.k12.in.us/physics/phynet/mechanics/circular%20motion/banked_no_friction.htm
Well, I have to say, this racetrack really knows how to make a smooth ride! Let's take a look at the situation.
When the car is traveling at a constant speed with no friction needed, it means that the normal force is equal to the car's weight. So, we can start by setting up some equations.
First, let's determine the forces acting on the car when it's on the banked track. We have the weight of the car (mg) acting downwards and a normal force (N) acting perpendicular to the track.
The vertical component of the normal force (N_y) provides the centripetal force needed to keep the car going in a circular motion. N_y is given by N_y = N * cos(theta).
The horizontal component of the normal force (N_x) balances out the car's weight and provides the necessary centripetal force. N_x is given by N_x = N * sin(theta).
Since we know that N_y = mg, where g is the acceleration due to gravity, we can rewrite that equation as N * cos(theta) = mg.
Similarly, we can write N_x = N * sin(theta) = mw^2 * r, where w is the angular velocity and r is the radius of the track.
Since we already have the speed of the car (28.0 m/s) and the radius of the track (390 m), we can find the angular velocity using the formula w = v / r: w = 28.0 m/s / 390 m = 0.0718 rad/s.
Now, let's substitute all the known values into the equations:
N * cos(theta) = mg
N * sin(theta) = mw^2 * r
After dividing the two equations, we get:
tan(theta) = w^2 * r / g
Plugging in the known values:
tan(theta) = (0.0718 rad/s)^2 * 390 m / (9.8 m/s^2)
And finally, solving for theta:
theta = arctan((0.0718^2 * 390) / 9.8)
Calculating that out gives us:
theta ≈ 0.167 radians
So, the angle (theta) at which the racetrack is banked is approximately 0.167 radians.
To determine the angle theta at which the racetrack is banked, we can use the concept of static friction and centripetal force.
The static friction force provides the necessary centripetal force to keep the car moving in a circular path without slipping. The static friction force can be calculated using the equation:
f_s = m * g * sin(theta)
where:
f_s is the static friction force,
m is the mass of the car,
g is the acceleration due to gravity, and
theta is the angle of the banked track.
At the maximum speed of 28.0 m/s, the static friction force is equal to zero since no friction is required. Therefore, we can set up the equation:
0 = m * g * sin(theta)
Since the mass (m) and acceleration due to gravity (g) are both positive constants, the sin(theta) term must be zero. This implies that theta must be equal to:
theta = arcsin(0)
theta = 0 degrees
Therefore, the racetrack should be banked at an angle of 0 degrees.
To determine theta, we can use the formula for the banking angle of a circular track:
tan(theta) = (v^2)/(g*r)
Where:
- theta is the banking angle
- v is the velocity of the car
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- r is the radius of the track
Step 1: Plug in the given values into the formula
tan(theta) = (28.0^2)/(9.8*390)
Step 2: Calculate the value inside the parentheses
tan(theta) = 784/3822
Step 3: Calculate the inverse tangent (tan^-1) of both sides to find theta
theta = tan^-1(784/3822)
Step 4: Use a calculator or trigonometric table to find the value of theta
theta ≈ 11.81 degrees (rounded to two decimal places)
Therefore, the angle theta is approximately 11.81 degrees.