A Banked Turn With Friction- min and max velocity

In summary, the conversation discusses the calculation of the maximum and minimum speeds a car can safely drive through a turn with a 25º angled bank, a radius of 50m, and a coefficient of static friction of 0.35. The formulas for calculating these speeds are F_net= Nsinθ + fcosθ= (tanθ+µ / 1-tanθ)mg and F_net= Nsinθ - fcosθ= (tanθ-µ / 1-tanθ)mg, and the resulting speeds are 21.6 m/s and 8.25 m/s, respectively. The asker requests for a more detailed derivation of these formulas.
  • #1
mmoadi
157
0

Homework Statement



A car drives through a turn with a bank angled at 25º according to horizontal floor. How much does the highest speed of the car have to be to prevent the car from slipping on the slope upwards? How much does the smallest speed of the car have to be to prevent the car from slipping on the slope downwards? Radius of the turn is 50m and the coefficient of static friction is 0, 35.

Homework Equations



F= mg
f= µN
F_centripetial= mv² / r

The Attempt at a Solution



For the max speed (v_max) that you can safely use for car to not slip up:

F_net= Nsinθ + fcosθ= (tanθ+µ / 1-tanθ)mg

F_net = F_centripetal
v_max= sqrt((tanθ+µ / 1-tanθ)rg)= 21.6 m/s

For the min speed (v_min) that you can safely use for car not to slip down:

F_net= Nsinθ - fcosθ= (tanθ-µ / 1-tanθ)mg

F_net = F_centripetal
v_max= sqrt((tanθ-µ / 1-tanθ)rg)= 8.25 m/s

Are my calculations correct?
 
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  • #2
Show your derivation a bit more detailed. Your formulas are not correct now. What do you mean on

(tanθ+µ / 1-tanθ) ? tanθ cancels.

ehild
ehild
 
  • #3
I would really like to know if my approach resolving this problem is correct?
thank you!
 

FAQ: A Banked Turn With Friction- min and max velocity

1. What is a banked turn with friction and how does it work?

A banked turn with friction is a type of turn that occurs when a vehicle or object moves along a curved path while maintaining a constant speed. The turn is banked at an angle, which helps to counteract the centrifugal force that would otherwise cause the object to slide off the path. Friction also plays a role in keeping the object on the path by providing a force that opposes the centrifugal force.

2. What is the minimum velocity required for a banked turn with friction?

The minimum velocity required for a banked turn with friction depends on several factors, including the angle of the bank, the radius of the turn, and the coefficient of friction between the object and the surface. In general, a higher bank angle and a smaller radius of turn will require a higher minimum velocity.

3. How is the maximum velocity determined for a banked turn with friction?

The maximum velocity for a banked turn with friction is determined by the balance between the centrifugal force and the frictional force. At a certain velocity, the centrifugal force becomes greater than the frictional force, causing the object to slide off the path. This maximum velocity can be calculated using the equation v = √(rgtanθ), where r is the radius of the turn, g is the acceleration due to gravity, and θ is the angle of the bank.

4. What happens if the coefficient of friction is increased in a banked turn?

If the coefficient of friction is increased, the maximum velocity for a banked turn will also increase. This is because a higher coefficient of friction means that there is a greater force opposing the centrifugal force, allowing for a higher maximum velocity before the object slides off the path.

5. How does the angle of the bank affect the minimum and maximum velocities in a banked turn with friction?

The angle of the bank has a significant impact on both the minimum and maximum velocities in a banked turn with friction. A higher bank angle will require a higher minimum velocity and can also increase the maximum velocity. However, if the bank angle is too high, it can cause the object to tip over or lose contact with the surface. Therefore, the angle of the bank must be carefully chosen to ensure safe and efficient turns.

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