A car of mass = 1200 traveling at 60.0 enters a banked turn covered with ice. The road is banked at an angle , and there is no friction between the road and the car's tires.What is the radius of the turn if = 20.0 (assuming the car continues in uniform circular motion around the turn)?
To find the radius of the turn, we can use the centripetal force formula. The centripetal force is provided by the horizontal component of the normal force on the car.
Step 1: Draw a free body diagram of the car on the banked turn:
N
↖ ↑ ↗
│ │ │
│ │ │
│ │ │
mg ←──┘ │ │ ←── centripetal force
Step 2: Analyze the forces acting on the car:
Vertical forces:
- The weight of the car (mg) pointing downwards
- The vertical component of the normal force (Nsinθ) pointing upwards
Horizontal forces:
- The centripetal force (Ncosθ) pointing inward toward the center of the turn
Step 3: Write the equations for the forces acting on the car:
In the vertical direction:
Nsinθ = mg
In the horizontal direction:
Ncosθ = mv²/r
Step 4: Solve the equations:
From the equation Nsinθ = mg, we can solve for N:
N = mg/sinθ
Substitute N in the horizontal equation:
(mg/sinθ)cosθ = mv²/r
Simplify:
gcosθ = v²/r
Step 5: Solve for the radius r:
r = v²/(gcosθ)
Given:
- Mass of the car (m) = 1200 kg
- Velocity of the car (v) = 60.0 m/s
- Angle of the banked turn (θ) = 20.0°
- Acceleration due to gravity (g) = 9.8 m/s²
Substitute the given values into the equation:
r = (60.0 m/s)² / (9.8 m/s²)(cos20.0°)
Calculating:
r = 219.18 m
Therefore, the radius of the turn is approximately 219.18 meters.
To find the radius of the turn, we can use the concept of centripetal force.
In this scenario, the only force acting on the car is the gravitational force (mg), which has two components: one perpendicular to the surface (mgcosθ), and the other along the surface (mgsinθ). The component along the surface provides the centripetal force required for uniform circular motion.
Given that there is no friction between the road and the car's tires, the centripetal force comes solely from the component of the gravitational force along the surface.
The formula for centripetal force is given by:
F = m * (V^2 / R)
Where:
F is the centripetal force,
m is the mass of the car,
V is the velocity of the car,
R is the radius of the turn.
Solving the equation for R, we get:
R = m * (V^2 / F)
Now, substituting the given values:
m = 1200 kg (mass of the car)
V = 60.0 m/s (velocity of the car)
θ = 20.0 degrees (angle of the banked turn)
First, we need to find the centripetal force F using the gravitational force along the surface:
F = m * g * sin(θ)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
F = 1200 kg * 9.8 m/s^2 * sin(20 degrees)
Calculating F, we find:
F ≈ 4104.4 N
Now, we can plug in the values to find the radius R:
R = 1200 kg * (60.0 m/s)^2 / 4104.4 N
Calculating R, we get:
R ≈ 1051.2 m
Therefore, the radius of the turn is approximately 1051.2 meters.