A curve of radius 59.9 m is banked so that a car of mass 1.4 Mg traveling with uniform speed 50 km/hr can round the curve without relying on friction to keep it from slipping on the surface.

At what angle is the curve banked? The acceleration due to gravity is 9.8 m/s2

* I swear I am doing this correctly I had it all set up and I keep getting 18.85 degrees but it says its wrong

50/km/h=13.9 m/s

m•a=N•sinα
0=N•cosα - m•g
m•v²/R=N•sinα = m•g•sin α/cos α =m•g•tan α

tan α = v²/gR =13.9²/9.8•59.9=0.329
α = tan⁻¹=18.19⁰

Well, it seems like you might need a bit of comedic relief after struggling with this question. Let me help you out, and hopefully, you'll find your answer and a reason to smile:

Ah, the wonderful world of physics and banking angles! Let's see if we can put a humorous spin on it.

Imagine this: a curve that's so scandalously inclined that even the cars feel like they're on a wild roller coaster ride. They're hanging on for dear life and hoping not to slip away!

Now, back to reality. We have a car of mass 1.4 Mg (that's quite heavy, I must say) zipping around this curve at a speed of 50 km/hr. To keep it from slipping, we need the magical power of centripetal force to come into play.

The centripetal force, in this case, comes from the component of the car's weight that's directed towards the center of the curve. To determine the banking angle, we can use this formula:

tan(θ) = (v^2) / (g * r)

Where:
- θ is the angle of the banked curve (what we're looking for)
- v is the speed of the car (50 km/hr, but we need it in m/s, so convert accordingly)
- g is the acceleration due to gravity (9.8 m/s^2)
- r is the radius of the curve (59.9 m)

Now, let's crunch those numbers and reach for the answer without any slipping or sliding comedy routines!

After plugging in the values, we find:

tan(θ) = (50^2) / (9.8 * 59.9)

tan(θ) ≈ 4.097

Now, to solve for θ, we need to take the arctan of both sides:

θ ≈ arctan(4.097)

And the moment you've been waiting for...

θ ≈ 76.43 degrees

So, there you have it! The angle of the curve should be approximately 76.43 degrees. Hopefully, this slightly amusing explanation was able to bring a smile to your face, even if the initial guess of 18.85 degrees didn't quite round the curve correctly. Keep up the great work, and remember: laughter is the best medicine, especially when it comes to tackling those mind-boggling physics problems!

To find the angle at which the curve is banked, we can use the concept of centripetal force. The centripetal force is provided by the component of the car's weight that acts perpendicular to the surface of the road.

Let's break down the problem step-by-step:

1. Convert the mass of the car from Mg to kg:
1.4 Mg = 1.4 × 10^3 kg

2. Convert the speed of the car from km/hr to m/s:
50 km/hr = (50 × 1000) / (60 × 60) m/s
50 km/hr = 13.89 m/s

3. Calculate the centripetal force using the formula:
centripetal force = mass × acceleration

Given:
mass (m) = 1.4 × 10^3 kg
acceleration (a) = v^2 / r (where v is the speed and r is the radius of the curve)

Substituting the values:
centripetal force = (1.4 × 10^3 kg) × (13.89 m/s)^2 / 59.9 m
centripetal force = 1.4 × 13.89^2 / 59.9 N

4. Calculate the component of the weight (mg) that acts perpendicular to the road, which provides the centripetal force:

The weight (mg) can be broken down into two components:
- m × g × sinθ (opposite the angle of banking)
- m × g × cosθ (perpendicular to the angle of banking)

Since the car is not relying on friction, the component of the weight perpendicular to the road provides the centripetal force:
m × g × cosθ = centripetal force

Substituting the values:
1.4 × 9.8 × cosθ = 1.4 × 13.89^2 / 59.9

5. Solve for cosθ:
cosθ = (1.4 × 13.89^2) / (59.9 × 1.4 × 9.8)

6. Find the angle θ using the inverse cosine (cos^-1) function:
θ = cos^-1 [(1.4 × 13.89^2) / (59.9 × 1.4 × 9.8)]

Calculating the above expression should give you the correct angle at which the curve is banked. Note that while the decimal value of 18.85 degrees may be close, it might not be exact due to rounding errors.

To find the angle at which the curve is banked, we can start by considering the forces acting on the car as it moves around the curve.

Since the car is traveling with uniform speed, we know that the net force acting on the car must be zero. This means that the centripetal force, which is the force that enables the car to move in a circular path, should be equal to the gravitational force acting on the car.

Let's break down the forces acting on the car:

1. Gravitational force (Fg):

The gravitational force acting on the car is equal to the product of its mass and the acceleration due to gravity:
Fg = m * g

Given:
Mass of the car (m) = 1.4 Mg = 1.4 * 1000 kg (since 1 Mg = 1000 kg)
Acceleration due to gravity (g) = 9.8 m/s²

Fg = (1.4 * 1000 kg) * 9.8 m/s² = 13,720 N

2. Centripetal force (Fc):

The centripetal force required to keep the car moving in a circular path is given by:
Fc = (m * v²) / r

Given:
Radius of the curve (r) = 59.9 m
Speed of the car (v) = 50 km/hr = 50 * 1000 / 3600 m/s (converting km/hr to m/s)

Fc = (1.4 * 1000 kg) * ((50 * 1000 / 3600) m/s)² / 59.9 m
Fc = 2,916.67 N

To keep the car from slipping, the downward component of the centripetal force should be balanced by the gravitational force acting perpendicular to the incline. This is achieved by the normal force (Fn) between the car and the inclined surface, given by:

Fn = Fg * cosθ
(θ is the angle of inclination)

We can determine Fn by substituting the known values into the equation:
Fn = 13,720 N * cosθ

Since the road is inclined, we can also consider the vertical component of the centripetal force, given by:

Fv = Fc * sinθ
Fv = 2,916.67 N * sinθ

Since the net force acting on the car must be zero and there is no horizontal acceleration, we have:

Fn - Fv = 0
13,720 N * cosθ - 2,916.67 N * sinθ = 0

Solving this equation for θ will give us the angle at which the curve is banked.

Now, let's solve for θ:

13,720 N * cosθ = 2,916.67 N * sinθ

Divide both sides by cosθ:
13,720 N = 2,916.67 N * tanθ

Divide both sides by 2,916.67 N:
tanθ = 13,720 N / 2,916.67 N
tanθ = 4.7055

To find θ, we can take the inverse tangent (arctan) of both sides of the equation:
θ = arctan(4.7055)

Calculating this angle will give you the correct answer.