27. A car travels along a horizontal road which is an
arc of a circle of radius 125m. the greatest speed.
at which the car can travel without slipping is
42km/hr. Find the coefficient of friction between
the tyres of the car and the road surface.
To start solving this problem, we need to use the formula for centripetal force:
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 circle
Since the car is not slipping, the frictional force provides the necessary centripetal force. Therefore, we can write:
F_friction = m * (v^2 / r)
The frictional force can also be written as:
F_friction = μ * N
Where:
μ is the coefficient of friction
N is the normal force
The normal force, N, is equal to the weight of the car, so we can write:
N = m * g
Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Substituting these equations into our original equation, we get:
μ * m * g = m * (v^2 / r)
Simplifying further:
μ * g = v^2 / r
Now we can plug in the given values:
g ≈ 9.8 m/s^2
v = 42 km/hr = (42 * 1000) / 3600 m/s (converting km/hr to m/s)
r = 125 m
Calculating further:
μ * 9.8 = ((42 * 1000) / 3600)^2 / 125
μ * 9.8 = 441 / 125
μ = (441 / 125) * (1 / 9.8)
μ ≈ 0.36
Therefore, the coefficient of friction between the tires of the car and the road surface is approximately 0.36.
To find the coefficient of friction between the tires of the car and the road surface, we need to consider the forces acting on the car while it is making a turn.
1. Identify the known values:
- Radius of the road, r = 125 m
- Maximum speed without slipping, v = 42 km/hr = 11.67 m/s
2. Calculate the acceleration of the car:
The maximum speed is the tangential speed of the car. Using the formula v = ωr, where ω is the angular velocity, we can find the angular velocity:
v = ωr
11.67 = ω * 125
ω = 0.09336 rad/s
Now, using the formula for centripetal acceleration, a = ω^2r, we can calculate the acceleration:
a = (0.09336)^2 * 125
a ≈ 1.17 m/s^2
3. Calculate the net force acting on the car:
The net force is the centripetal force required to keep the car moving in a circle. Using the formula F = ma, the net force can be calculated:
F = m * a
4. Calculate the normal force:
The normal force is the force exerted by the ground on the car and is equal to the weight of the car due to the absence of vertical acceleration. It can be calculated as:
N = m * g
5. Calculate the frictional force:
The frictional force is the force that opposes the motion of the car. It can be calculated as:
f = μ * N
6. Set up an equation for the net force:
Since the net force is equal to the frictional force, we can write:
F = f
7. Substitute the values and solve for the coefficient of friction:
m * a = μ * m * g
a = μ * g
μ = a / g
μ = 1.17 / 9.81
μ ≈ 0.12
Therefore, the coefficient of friction between the tires of the car and the road surface is approximately 0.12.