A flat (unbanked) curve on a highway has a radius of 220m. A car rounds the curve at a speed of 25m/s. What is the minimum coefficient of friction that will prevent sliding?
A physics book slides off a table with a horizontal speed of 1.10m/s . It strikes the floor in 0.350s. Ignore air resistance.
Determine:
a. The velocity just before it hits the floor.
b. The height of the table top above the floor.
c. The horizontal distance from the edge of the table to the point where the book strikes the floor.
To find the minimum coefficient of friction that will prevent sliding, we need to consider the centripetal force acting on the car as it goes around the curve.
The centripetal force is provided by the friction between the tires and the road. If the friction force is insufficient, the car will slide outwards.
The formula for centripetal force is:
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, and
r is the radius of the curve.
In this case, we have the radius of the curve (r = 220m) and the velocity of the car (v = 25m/s). We can assume the mass of the car (m) cancels out when calculating the coefficient of friction (μ), so we don't need to know the mass.
The formula for the frictional force is:
F_friction = μ * N
Where:
F_friction is the frictional force,
μ is the coefficient of friction, and
N is the normal force.
To find the minimum coefficient of friction, we can equate the centripetal force to the frictional force:
F = F_friction
(m * v^2) / r = μ * N
Since the car is on a flat road, the normal force N is equal to the weight of the car, which is given by:
N = m * g
Where g is the acceleration due to gravity. Again, the mass cancels out.
Substituting the expression for N into the equation for F_friction, we get:
(m * v^2) / r = μ * (m * g)
Canceling out the mass (m) from both sides of the equation, we have:
(v^2) / r = μ * g
Rearranging the equation, we can solve for the coefficient of friction (μ):
μ = (v^2) / (g * r)
Now we can plug in the known values to find the minimum coefficient of friction:
μ = (25^2) / (9.8 * 220)
Calculating further:
μ = 625 / (9.8 * 220)
μ ≈ 0.285
So, the minimum coefficient of friction that will prevent sliding on this curve is approximately 0.285.
To prevent sliding, friction must provide (i.e. equal) the centripetal force
μmg=mv²/r
Solve for μ, with
v=25 m/s
r=220 m
g=9.8 m/s²
m cancels