A car rounds a banked curve where the radius of curvature of the road is R=100m., the banking angle 10 degrees and the coefficient of static friction is .10 (slippery conditions). A) Determine the range of speeds the car can gave without slipping up or down the road.
down slip problem:
forces along slope pointed down slope:
m g sin 10 - .1 m g cos 10 = m (v^2/r)cos 10
up slip problem, reverse direction of friction
m g sin 10 +.1 m g cos 10 = m (v^2/r) cos 10
note m cancels everywhere
4.5M/S
To determine the range of speeds the car can have without slipping up or down the road, we need to consider the following:
1. The maximum static frictional force that can be applied by the road on the car without slipping is given by:
f_max = μ_s * m * g
where:
μ_s = coefficient of static friction
m = mass of the car
g = acceleration due to gravity (~9.8 m/s^2)
2. The components of the gravitational force acting on the car parallel and perpendicular to the road are:
F_g_parallel = m * g * sin(θ)
F_g_perpendicular = m * g * cos(θ)
where:
θ = banking angle
3. The net force acting on the car along the radial direction is:
F_net = f_max - F_g_parallel
Since the car is not slipping up or down the road, the net force must provide the necessary centripetal force:
F_net = m * (v^2 / R)
where:
v = velocity of the car
4. Combining equations 3 and 4, we can determine the range of speeds:
m * (v^2 / R) = f_max - F_g_parallel
Now, let's calculate the range of speeds in steps:
Step 1: Calculate the maximum static frictional force.
f_max = μ_s * m * g
= 0.10 * m * 9.8
Step 2: Calculate the components of the gravitational force.
F_g_parallel = m * g * sin(θ)
= m * 9.8 * sin(10°)
F_g_perpendicular = m * g * cos(θ)
= m * 9.8 * cos(10°)
Step 3: Calculate the net force.
F_net = f_max - F_g_parallel
Step 4: Equate the net force to the centripetal force.
m * (v^2 / R) = F_net
Step 5: Solve for the range of speeds.
v^2 = R * (F_net / m)
v = sqrt(R * (F_net / m))
To determine the range of speeds, we need the given mass of the car. Please provide the mass of the car.
To determine the range of speeds the car can have without slipping up or down the road, we need to consider the forces acting on the car.
First, let's define the variables:
R = radius of curvature of the road = 100m
θ = banking angle = 10 degrees
μs = coefficient of static friction = 0.10
The forces acting on the car are gravity (mg), the normal force (N), and the friction force (f).
The normal force N can be resolved into two components:
N_parallel = N * cos(θ) (the component acting parallel to the road)
N_perpendicular = N * sin(θ) (the component acting perpendicular to the road)
The maximum static friction force that can act on the car is given by:
f_max = μs * N_parallel
Since the car is not slipping, the static friction force provides the centripetal force required for circular motion:
f = mv^2 / R
Now, let's set up the equations based on the given information:
1) Equating the forces on the car parallel to the road:
f_max = mv^2 / R
Substituting the expression for N_parallel:
μs * N * cos(θ) = m * v^2 / R
2) Equating the forces on the car perpendicular to the road:
mg - N_perpendicular = 0
mg - N * sin(θ) = 0
N = mg / sin(θ)
Now, we can substitute the expression for N in equation 1:
μs * (mg / sin(θ)) * cos(θ) = m * v^2 / R
Simplifying the equation:
μs * g * cos(θ) / sin(θ) = v^2 / R
Since g, μs, θ, and R are known constants, we can rearrange the equation to solve for v:
v^2 = μs * g * R * cos(θ) / sin(θ)
Taking the square root of both sides:
v = sqrt(μs * g * R * cos(θ) / sin(θ))
Now, we can calculate the range of speeds by considering the range of possible values for θ (banking angle).
In this case, the car is rounding a positively banked curve (θ = 10 degrees). The range of speeds can be calculated by varying the coefficient of static friction μs:
For the car not to slip up:
v_up = sqrt(μs * g * R * cos(θ) / sin(θ))
For the car not to slip down:
v_down = sqrt(μs * g * R * cos(θ) / sin(θ))
Substituting the values:
v_up = sqrt(0.10 * 9.8 m/s^2 * 100 m * cos(10 degrees) / sin(10 degrees))
v_down = sqrt(0.10 * 9.8 m/s^2 * 100 m * cos(10 degrees) / sin(10 degrees))
Calculating the values:
v_up = 9.95 m/s
v_down = 5.93 m/s
Therefore, the range of speeds the car can have without slipping up or down the road is between approximately 5.93 m/s and 9.95 m/s.