A turn of radius 20m is banked for the vehicles going at a speed of 36km/h. If the coefficient of static friction between road and the tyre is 0.4, what are the possible speeds of a vehicle so that it neither slips down nor skids up ?

13 km/h as friction oposses the velocity to stop skidding.

To determine the possible speeds at which the vehicle neither slips down nor skids up on a banked turn, we need to consider the forces acting on the vehicle.

In this scenario, the gravitational force, the normal force, and the frictional force are the relevant forces.

1. The gravitational force (Fg) can be calculated using the formula:
Fg = m * g
where m is the mass of the vehicle and g is the acceleration due to gravity (approximately 9.8 m/s^2).

2. The normal force (Fn) can be calculated using the following equation for a banked turn:
Fn = m * g * cos(θ)
where θ is the angle of the incline or banking angle.

3. The frictional force (Ff) can be calculated using:
Ff = μs * Fn
where μs is the coefficient of static friction between the road and the tire.

4. The centripetal force (Fc) acting towards the center of the turn can be calculated using:
Fc = m * v^2 / r
where v is the velocity of the vehicle and r is the radius of the turn.

To ensure the vehicle neither slips down nor skids up, the frictional force should be enough to provide the necessary centripetal force. Therefore, we equate the two:

Fc = Ff

Substituting the relevant equations:

m * v^2 / r = μs * m * g * cos(θ)

Cancelling mass (m) from both sides of the equation:

v^2 / r = μs * g * cos(θ)

Rearranging the equation to find the velocity (v):

v^2 = μs * g * r * cos(θ)

v = √(μs * g * r * cos(θ))

Plugging in the given values:

v = √(0.4 * 9.8 * 20 * cos(θ))

Now, we can calculate the possible speeds for different angles of incline (θ).

To determine the possible speeds of a vehicle on a banked turn so that neither slipping nor skidding occurs, we need to consider the forces acting on the vehicle.

Firstly, let's find the angle of banking for the turn. The angle of banking (θ) can be calculated using the following formula:

θ = arctan(v^2 / (g * r))

where:
- θ is the angle of banking,
- v is the velocity of the vehicle,
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- r is the radius of the turn.

Given that the radius of the turn (r) is 20m, we can calculate θ:

θ = arctan((36 km/h)^2 / (9.8 m/s^2 * 20m))

Next, we need to understand the forces acting on the vehicle on the banked turn:

1. Gravity force (mg): This force acts vertically downward and can be broken down into two components:
- Perpendicular to the road surface: mg * cos(θ)
- Parallel to the road surface: mg * sin(θ)

2. Frictional force (f): This force acts horizontally and opposes the sliding motion. It can be calculated using the formula:
f = μ * N
where μ is the coefficient of static friction and N is the normal force acting on the vehicle.

To neither slip down nor skid up, the combined forces in the horizontal direction should provide the necessary centripetal force for circular motion:

f = m * v^2 / r

Now let's calculate the possible speeds for the vehicle, considering these forces:

1. To prevent slipping down:
In this case, the frictional force provides the necessary centripetal force:
f = m * v^2 / r

2. To prevent skidding up:
In this case, the frictional force is not enough to provide the necessary centripetal force, so the maximum frictional force is used:
f_max = μ * N_max
N_max = mg * cos(θ)
f_max = μ * mg * cos(θ)

The remaining centripetal force needed is provided by the horizontal component of the weight:
f_remaining = m * v^2 / r - f_max

Finally, we can calculate the speeds at which neither slipping down nor skidding up occurs:

1. To prevent slipping down:
Set the frictional force equal to the centripetal force:
m * v^2 / r = μ * mg * sin(θ)
Solve for v.

2. To prevent skidding up:
Set the remaining centripetal force equal to the horizontal component of the weight:
m * v^2 / r - f_max = mg * sin(θ)
Solve for v.

Substitute the values of μ, m, g, θ, and r into the equations and solve to find the possible speeds in each case.