A race car enters a flat 200 m radius curve at a speed of 20 m/swhile increasing its speed at a constant 2 m/s2. If thecoefficient of static friction is .700, what will the speed of thecar be when the car beings to slide?
To determine the speed at which the car begins to slide, we need to find the maximum speed at which the static frictional force can counteract the centrifugal force.
Step 1: Determine the maximum static frictional force (F_static_max).
The maximum static frictional force is given by the formula:
F_static_max = coefficient of static friction * normal force
The normal force (N) is equal to the weight (W) of the car, which is the mass (m) of the car multiplied by the acceleration due to gravity (g).
Given that the coefficient of static friction is 0.700, we can plug in the values:
F_static_max = 0.700 * (m * g)
Step 2: Calculate the centrifugal force (F_centrifugal).
The centrifugal force acting on the car is given by the formula:
F_centrifugal = m * (v^2 / r)
Where:
- m is the mass of the car,
- v is the velocity of the car,
- r is the radius of the curve.
Given that the car enters the curve with a speed of 20 m/s and a radius of 200 m, we can plug in the values:
F_centrifugal = m * (20^2 / 200)
Step 3: Equate the maximum static frictional force and the centrifugal force, and solve for the velocity (v_slide) at which the car begins to slide.
F_static_max = F_centrifugal
0.700 * (m * g) = m * (20^2 / 200)
Step 4: Simplify the equation and solve for v_slide.
0.700 * g = 20^2 / 200
v_slide = sqrt((0.700 * g * 200) / 20^2)
To get the numerical value, substitute the acceleration due to gravity (g) as 9.8 m/s^2:
v_slide = sqrt((0.700 * 9.8 * 200) / 20^2) ≈ 14.83 m/s
Therefore, the speed of the car when it begins to slide will be approximately 14.83 m/s.
To find the speed at which the car begins to slide, we need to determine the maximum speed that can be achieved before the frictional force is overcome.
The first step is to calculate the maximum frictional force that can act on the car. We can use the equation:
F_friction = μ * N
where F_friction is the frictional force, μ is the coefficient of static friction, and N is the normal force.
The normal force can be calculated using the equation:
N = m * g
where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's calculate the maximum frictional force:
N = m * g
= m * 9.8
F_friction = μ * N
= μ * m * 9.8
Next, we need to determine the net force acting on the car in the radial direction. It is given by the equation:
F_net = m * a
where a is the acceleration of the car. In this case, the acceleration is due to the change in speed and can be calculated using the equation:
a = v^2 / r
where v is the velocity of the car and r is the radius of the curve.
Now, let's calculate the net force:
F_net = m * a
= m * (v^2 / r)
To keep the car on the curve, the net force should be equal to or less than the maximum static frictional force. Therefore, we can write the equation:
F_net ≤ F_friction
Substituting the values, we get:
m * (v^2 / r) ≤ μ * m * 9.8
Simplifying, we find:
v^2 / r ≤ μ * 9.8
Now, we can rearrange the equation to solve for v:
v^2 ≤ μ * 9.8 * r
Taking the square root of both sides, we have:
v ≤ √(μ * 9.8 * r)
Let's plug in the values:
v ≤ √(0.7 * 9.8 * 200)
Calculating, we find:
v ≤ √(134.68)
v ≤ 11.6 m/s
Therefore, the speed of the car when it begins to slide is approximately 11.6 m/s.