2. Determine the velocity that a car should have while traveling around a frictionless curve of radius 100m and that is banked 20 degrees.
m * centripetal acceleration inward = m v^2/R
force up = mg
total force normal to surface so no slip
tan 20 = (v^2 /R)/g = v^2 / (R g)
Why did the car bring a blanket to the frictionless curve? Because it wanted to have a comfortable slide! Now, let's calculate the velocity this car should have while cruising around that curve. To do that, we can use the formula for the critical velocity on a banked curve:
v = sqrt(r * g * tan(theta))
where:
v is the velocity,
r is the radius of the curve (100m in this case),
g is the acceleration due to gravity (approximately 9.8 m/s^2), and
theta is the bank angle (20 degrees).
Plugging in the values, we get:
v = sqrt(100 * 9.8 * tan(20))
v = sqrt(980 * tan(20))
v ≈ sqrt(980 * 0.36397)
v ≈ sqrt(357.1706)
v ≈ 18.89 m/s
So, the car should have a velocity of approximately 18.89 m/s to cruise smoothly around this frictionless, banked curve. And don't forget the blanket for a comfy slide!
To determine the velocity that a car should have while traveling around a frictionless curve, we can use the concept of centripetal force and the relationship between the velocity, radius, and angle of the banked curve.
The formula for the centripetal force is:
F_c = (m * v^2) / r
Where:
F_c is the centripetal force
m is the mass of the car
v is the velocity of the car
r is the radius of the curve
In a banked curve, the vertical component of the Normal force (Nsinθ) provides the necessary centripetal force. Here, θ is the angle of banking.
Since the curve is frictionless, there is no horizontal force to consider.
The vertical component of the Normal force is given by:
Nsinθ = (m * v^2) / r
The Normal force (N) can be expressed in terms of the gravitational force (mg) and the angle of banking (θ) as:
N = mg / cosθ
Substituting this value of N in the equation above, we get:
(mg / cosθ) * sinθ = (m * v^2) / r
Simplifying further:
g * tanθ = (v^2) / r
Now we can solve for the velocity (v). Rearranging the equation:
v^2 = g * r * tanθ
Taking the square root of both sides:
v = √(g * r * tanθ)
Given that the radius (r) is 100m and the angle of banking (θ) is 20 degrees, we can substitute these values into the equation to determine the velocity:
v = √(9.8 m/s^2 * 100m * tan20°)
Calculating this, we find:
v ≈ 17.9 m/s
So, the car should have a velocity of approximately 17.9 m/s while traveling around the frictionless curve of radius 100m and being banked at 20 degrees.
To determine the velocity that a car should have while traveling around a frictionless curve, we can use the concept of centripetal force.
The centripetal force required to keep an object moving in a circular path can be calculated using the formula:
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, and r is the radius of the circular path.
In this case, we are given the radius of the curve (r = 100m) and the banking angle (20 degrees). Since the curve is frictionless, there is no need to consider any friction force.
Now, let's solve for the velocity (v).
Step 1: Convert the banking angle from degrees to radians.
The formula for converting degrees to radians is:
radians = (pi/180) * degrees
In this case, the banking angle in radians is:
20 degrees * (pi/180) = 0.35 radians
Step 2: Determine the gravitational force component that provides the necessary centripetal force.
When a curve is banked, there are two forces at play - the gravitational force component perpendicular to the curve (mg * sin(theta)) and the normal force (mg * cos(theta)). The net centripetal force comes from the gravitational force component perpendicular to the curve.
Step 3: Equate the gravitational force component to the centripetal force equation.
mg * sin(theta) = (m * v^2) / r
Step 4: Simplify the equation to solve for v.
mg * sin(theta) = (m * v^2) / r
Rearrange the equation to solve for v:
v^2 = r * g * sin(theta)
v = sqrt(r * g * sin(theta))
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 5: Substitute the given values into the equation to find the velocity.
v = sqrt(100m * 9.8 m/s^2 * sin(0.35 radians))
v ≈ sqrt(980 m^2/s^2 * 0.342)
v ≈ sqrt(335.6)
v ≈ 18.32 m/s
Therefore, the car should have a velocity of approximately 18.32 m/s while traveling around the frictionless curve of radius 100m and being banked at 20 degrees.