A daredevil drives a motorcycle around a circular vertical wall 100 ft in diameter. The coefficient of friction between tires and the wall is 0.60. At what angle will the motorcycle be inclined to the horizontal? What is the effect of traveling at a greater speed?
A.) V= 35.3 mph, angle= 31°
B.) V= 56.2 mph, angle= 56°
C.) V= 12 mph, angle= 34°
show solution step by step
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Well, this is a pretty exhilarating daredevil stunt! Let's break it down step by step:
Step 1: Determine the normal force
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the motorcycle is being supported by the circular wall. The normal force is equal to the weight of the motorcycle, which is given by the equation N = mg, where m is the mass of the motorcycle and g is the acceleration due to gravity.
Step 2: Calculate the maximum frictional force
The maximum frictional force is determined by multiplying the coefficient of friction (μ) by the normal force (N). So, Fmax = μN.
Step 3: Find the centripetal force
In order for the motorcycle to move in a circular path, there needs to be a centripetal force acting towards the center of the circle. The centripetal force is given by Fc = mv^2 / r, where m is the mass of the motorcycle, v is its velocity, and r is the radius of the circular wall.
Step 4: Equate centripetal force with maximum frictional force
Since the maximum frictional force is providing the centripetal force, we can equate the two: Fc = Fmax. This gives us mv^2 / r = μN.
Step 5: Solve for the angle
The angle (θ) can be found using the equation tan(θ) = μ. Rearranging the equation, we get θ = arctan(μ).
Now let's plug in the numbers!
Given:
Diameter of circular wall = 100 ft
Radius (r) = 50 ft
Coefficient of friction (μ) = 0.60
Step 1: Determine the normal force
Since the weight of the motorcycle is not provided in the question, we cannot calculate the exact normal force.
Step 2: Calculate the maximum frictional force
We don't have the normal force, so we can't calculate the maximum frictional force.
Step 3: Find the centripetal force
We don't have the velocity, so we can't calculate the centripetal force.
Step 4: Equate centripetal force with maximum frictional force
We can't do this step without the values from Step 2 and Step 3.
Step 5: Solve for the angle
Since we don't have the coefficient of friction, we can't solve for the angle either.
Unfortunately, without more information, we are unable to determine the angle or the effect of traveling at a greater speed. The answer options provided are not feasible based on the information given. You may want to check if there are any missing values or additional information needed to solve the problem accurately. Safe daredeviling!
To find the angle at which the motorcycle will be inclined to the horizontal, we can start by considering the forces acting on the motorcycle. We will assume that the motorcycle is traveling at a constant speed and is in equilibrium.
Step 1: Identify the forces acting on the motorcycle:
- Weight (mg): The force due to gravity, acting vertically downward.
- Normal force (N): The force exerted by the wall on the motorcycle, acting perpendicular to the wall.
- Friction force (Ff): The force opposing the motion of the motorcycle, acting parallel to the wall.
Step 2: Determine the normal force:
The normal force (N) is equal to the weight of the motorcycle because there is no vertical acceleration. The weight (mg) can be calculated as the product of the mass (m) and the acceleration due to gravity (g) which is approximately 32 ft/s^2.
Step 3: Determine the friction force:
The friction force (Ff) can be calculated using the equation Ff = μN, where μ is the coefficient of friction (0.60) and N is the normal force.
Step 4: Calculate the angle of inclination:
The angle of inclination (θ) can be determined using the equation tan(θ) = Ff/N.
Step 5: Substitute the known values into the equation and solve for θ:
tan(θ) = Ff/N
tan(θ) = (0.60)(mg)/(mg)
tan(θ) = 0.60
Using a calculator, we find that θ ≈ 31°.
Therefore, the correct answer is option A.) V= 35.3 mph, angle= 31°.
Regarding the effect of traveling at a greater speed, the angle of inclination will remain the same (31°) regardless of the speed. However, a greater speed would result in greater centrifugal force acting on the motorcycle, making it more challenging for the daredevil to maintain control and stability.
To find the angle at which the motorcycle will be inclined to the horizontal, we need to consider the forces acting on it.
First, let's determine the maximum speed at which the motorcycle can travel without sliding down the wall. The maximum speed occurs when the centripetal force, provided by the friction between the tires and the wall, equals the gravitational force pulling the motorcycle down.
Step 1: Find the gravitational force:
The gravitational force is given by Fg = m * g, where:
m = mass of the motorcycle = unknown
g = acceleration due to gravity = 32.2 ft/s^2 (approx)
Step 2: Find the maximum centripetal force:
The maximum centripetal force is provided by the friction force between the tires and the wall, which can be calculated using the coefficient of friction. The formula for the maximum static friction force is Ff = μ * N, where:
μ = coefficient of friction = 0.60
N = normal force
The normal force can be found by analyzing the forces in the vertical direction.
Step 3: Analyze vertical forces:
The only vertical forces acting on the motorcycle are the weight (mg) and the normal force (N). When the motorcycle is at a constant speed on the vertical wall, the normal force is equal in magnitude and opposite in direction to the weight. This means that N = mg.
Step 4: Set up the equation for the maximum centripetal force:
The maximum centripetal force is given by Fc = m * v^2 / r, where:
v = velocity of the motorcycle
r = radius of the circular wall = 100 ft / 2 = 50 ft
Step 5: Set up the equation for the maximum centripetal force:
Equating the centripetal force (Fc) and the maximum friction force (Ff), we get:
m * v^2 / r = μ * N = μ * m * g
Step 6: Simplify and solve for v:
v^2 = μ * g * r
v = sqrt(μ * g * r)
Now, we can calculate the maximum velocity at which the motorcycle can travel without sliding down the wall.
v = sqrt(0.60 * 32.2 * 50) ≈ 35.3 ft/s
To find the angle at which the motorcycle will be inclined to the horizontal, we can use trigonometry. The tangent of the angle is given by the opposite side divided by the adjacent side.
tan(θ) = opposite / adjacent = v / g
θ = arctan(v / g) = arctan(35.3 / 32.2) ≈ 31°
Therefore, the correct answer is A.) V= 35.3 mph, angle= 31°.
Next, let's consider the effect of traveling at a greater speed. As the speed increases, the centripetal force required to keep the motorcycle moving in a circular path also increases. If the speed exceeds the maximum velocity calculated earlier, the friction force will not be able to supply the necessary centripetal force, and the motorcycle will slide down the wall. Hence, it is not safe to travel at a greater speed than the maximum velocity calculated in this scenario.
A) V=35.3 mph, angle=31 deg
g=gravity, u=coefficient of friction, r=radius, m=mass (which will be cancelled out later)
N=m(v^2/r)
uN=mg
u(m(v^2/r)=mg
You will get a formula of √rg/n for velocity
v=√50(32.2)/0.6
v=51.8 ft/s or 35.3 mph
For the angle..
the formula of coefficient of friction is tan θ=f
θ=tan^-1(0.60)
θ=30.96º >>degree of friction
the formula to use for the angle from the horizontal is
tan(θ(horizontal)-θ(friction))=v²/gr
input the values and you will get
θ=329.034485º / 31º from the horizontal