If the coefficient of static friction for tires on a road is 0.25, at what maximum speed can a car round a level 47.5 m radius curve without slipping?
10.787
To find the maximum speed at which a car can round a curve without slipping, we can use the concept of centripetal force.
The centripetal force required for an object to move in a curved path is given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the object
v is the velocity of the object
r is the radius of the curve
In this case, we want to find the maximum speed at which the car can go without slipping, which means the friction force must be equal to or greater than the centripetal force. The friction force is given by:
F friction = µ * m * g
Where:
µ is the coefficient of static friction (0.25 in this case)
m is the mass of the car
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Setting the friction force equal to the centripetal force, we can solve for the maximum speed:
µ * m * g = (m * v^2) / r
Simplifying the equation:
v^2 = µ * g * r
Now, let's plug in the given values:
µ = 0.25
r = 47.5 m
g = 9.8 m/s^2
v^2 = 0.25 * 9.8 * 47.5
Solving for v:
v = sqrt(0.25 * 9.8 * 47.5)
v ≈ sqrt(115.625)
v ≈ 10.76 m/s
Therefore, the maximum speed at which the car can round the curve without slipping is approximately 10.76 m/s.
To determine the maximum speed at which a car can round a curve without slipping, we need to consider the centripetal force and the frictional force. The maximum frictional force is equal to the product of the coefficient of static friction and the normal force.
In this case, the normal force is equal to the weight of the car, which is given by the equation:
Normal force = mass × gravitational acceleration
First, we need to calculate the normal force. Since the weight of the car is dependent on its mass and the acceleration due to gravity (9.8 m/s^2), we need the mass of the car. Let's assume the mass is 1000 kg.
Normal force = 1000 kg × 9.8 m/s^2
Normal force = 9800 N
Now, we can calculate the maximum frictional force:
Maximum frictional force = coefficient of static friction × normal force
Maximum frictional force = 0.25 × 9800 N
Maximum frictional force = 2450 N
The maximum frictional force provides the centripetal force needed to keep the car on the curve:
Centripetal force = mass × velocity^2 / radius
Since we are solving for maximum velocity, we can rearrange the formula:
Velocity^2 = (Centripetal force × radius) / mass
Velocity^2 = (Maximum frictional force × radius) / mass
Velocity^2 = (2450 N × 47.5 m) / 1000 kg
Velocity^2 = 116.375 m^2/s^2
Finally, we can solve for the maximum velocity:
Velocity = √(116.375 m^2/s^2)
Velocity ≈ 10.79 m/s
Therefore, the maximum speed at which the car can round a level 47.5 m radius curve without slipping is approximately 10.79 m/s.
The maximum speed possible, Vm, is that for which the centripetal force equals the maximum possible friction force
MVm^2/R = M*g*Us
where Us is the statc friction cefficient.
Vm = sqrt(g*R*Us)