Two banked curves have the same radius. Curve A is banked at an angle of 12°, and
curve B is banked at an angle of 20°. A car can travel around curve A without relying on
friction at a speed of 19.5 m/s. At what speed can this car travel around curve B without
relying on friction?
To determine the speed at which the car can travel around curve B without relying on friction, we can use the concept of equilibrium of forces.
First, let's analyze the forces acting on the car as it travels around the banked curve. In both cases, we have two main forces - the normal force (N) and the gravitational force (mg).
For curve A:
The normal force N can be resolved into two components: N_vertical acting perpendicular to the horizontal surface and N_horizontal acting parallel to the horizontal surface. The gravitational force mg acts vertically downwards.
Since the car can travel around curve A without relying on friction, the horizontal component of the normal force N_horizontal provides the necessary centripetal force to keep the car in circular motion.
For curve B:
Similarly, the normal force N can be resolved into N_vertical and N_horizontal components, and the gravitational force mg acts vertically downwards.
We know that the centripetal force required for circular motion is given by the equation:
Fc = m * v^2 / r
where Fc is the centripetal force, m is the mass of the car, v is the velocity, and r is the radius of the curve.
For both curves, the centripetal force is provided by the horizontal component of the normal force N_horizontal.
Let's compare the horizontal components of the normal forces for curves A and B:
For curve A:
N_horizontal_A = N * sin(θ_A) (since sin(θ) = opposite/hypotenuse)
For curve B:
N_horizontal_B = N * sin(θ_B)
Since both banked curves have the same radius, the gravitational force remains the same for both cases, and we can consider that m * g is constant.
For curve A:
N_horizontal_A = m * g * sin(θ_A)
For curve B:
N_horizontal_B = m * g * sin(θ_B)
Since we want to find the speed at which the car can travel around curve B without relying on friction, we can set the horizontal components of the normal forces for both curves equal to each other and solve for the velocity:
N_horizontal_A = N_horizontal_B
m * g * sin(θ_A) = m * g * sin(θ_B)
The mass (m) and the gravitational force (g) are common factors that cancel out, so we are left with:
sin(θ_A) = sin(θ_B)
Now, we can solve for the velocity:
v_B = sqrt((sin(θ_A) / sin(θ_B)) * v_A^2)
Substituting the known values:
θ_A = 12°
θ_B = 20°
v_A = 19.5 m/s
v_B = sqrt((sin(12°) / sin(20°)) * (19.5 m/s)^2)
Calculating this expression will give us the speed at which the car can travel around curve B without relying on friction.
To determine the speed at which the car can travel around curve B without relying on friction, we can use the concept of centripetal force.
The centripetal force is provided by the horizontal component of the car's weight. It can be calculated as:
Fc = (m * v^2) / r
Where:
- Fc is the centripetal force
- m is the mass of the car
- v is the velocity of the car
- r is the radius of the curve
Since both curves have the same radius, we can set the centripetal forces for curve A and curve B equal to each other:
(m * vA^2) / rA = (m * vB^2) / rB
We know:
- curve A is banked at an angle of 12°
- curve B is banked at an angle of 20°
- the velocity around curve A is 19.5 m/s
To determine the velocity around curve B, we need to find the relation between the angles and the radius.
For any banked curve, the relation between the angle and the radius can be given by:
tan(angle) = (v^2) / (g * r)
Where:
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Rearranging the equation, we can solve for r:
r = (v^2) / (g * tan(angle))
Now we substitute the values for curve A:
rA = (19.5^2) / (9.8 * tan(12°))
Calculate rA.
Next, we substitute the values for curve B:
rB = (19.5^2) / (9.8 * tan(20°))
Calculate rB.
Now we substitute the values of rA and rB into the equation we derived earlier:
(m * vA^2) / rA = (m * vB^2) / rB
Substitute the known values and solve for vB to determine the speed at which the car can travel around curve B without relying on friction.