A motorcycle is traveling up one side of a hill and down the other side. The crest of the hill is a circular arc with a radius of 68.0 m. Determine the maximum speed that the cycle can have while moving over the crest without losing contact with the road.
g = v^2/r
To determine the maximum speed that the motorcycle can have without losing contact with the road, we can use the concept of centripetal force.
At the crest of the hill, the motorcycle is subject to two forces: the gravitational force pulling it downward and the centripetal force keeping it in circular motion. The gravitational force can be split into two components: the vertical component (mgcosθ) and the horizontal component (mgsinθ), where θ is the angle of the hill with respect to the horizontal axis.
At the maximum speed, the centripetal force is equal to the horizontal component of the gravitational force. Therefore, we can set up the following equation:
mgsinθ = mv^2 / r
Where m represents the mass of the motorcycle, g is the acceleration due to gravity, v is the speed of the motorcycle, and r is the radius of the circular arc.
To find the maximum speed, we need to find the value of sinθ.
sinθ = opposite / hypotenuse = r / (r + h)
Where h is the height of the hill. Since the hill is described as a circular arc, the height of the hill can be found using the Pythagorean theorem:
h^2 = (2r)^2 - r^2
Simplifying this equation gives:
h^2 = 3r^2
Taking the square root of both sides gives:
h = sqrt(3) * r
Now we can substitute this value of h into the equation for sinθ:
sinθ = r / (r + sqrt(3) * r) = 1 / (1 + sqrt(3))
Substituting this value of sinθ into the equation for the maximum speed:
mgsinθ = mv^2 / r
mg(1 / (1 + sqrt(3))) = mv^2 / r
Simplifying this equation gives:
v^2 = rg(1 + sqrt(3))
Taking the square root of both sides gives the maximum speed:
v = sqrt(rg(1 + sqrt(3)))
Now we can substitute the values given. Assuming g is equal to 9.8 m/s^2 for the acceleration due to gravity:
v = sqrt((68.0 m) * (9.8 m/s^2) * (1 + sqrt(3)))
Calculating this gives:
v ≈ 9.8 m/s
Therefore, the motorcycle can have a maximum speed of approximately 9.8 m/s while moving over the crest without losing contact with the road.
To determine the maximum speed that the motorcycle can have while moving over the crest without losing contact with the road, we need to consider the forces acting on the motorcycle at the top of the hill.
At the top of the hill, the motorcycle experiences two forces: the downward force due to gravity (mg) and the upward force provided by the normal force of the road surface (N). For the motorcycle to stay in contact with the road, the normal force must be greater than or equal to zero.
At the top of the hill, the net force acting on the motorcycle is the centripetal force (Fc). The centripetal force is provided by the horizontal component of the normal force (Ncosθ), where θ is the angle between the normal force and the vertical direction.
We can determine the maximum speed by finding the point when the normal force is zero. At this point, the motorcycle is just about to lose contact with the road.
Using the following formula for the centripetal force at the top of the hill:
Fc = m * v^2 / r
where:
- Fc is the centripetal force
- m is the mass of the motorcycle
- v is the speed of the motorcycle
- r is the radius of the circular arc (in this case, 68.0 m)
At the top of the hill, the centripetal force is provided by the horizontal component of the normal force:
Fc = Ncosθ
Since the motorcycle is just about to lose contact with the road at this point, the normal force is zero:
Ncosθ = 0
Therefore, the maximum speed (vmax) can be found by rearranging the formula for centripetal force:
Fc = m * vmax^2 / r
Ncosθ = m * vmax^2 / r
We can solve for vmax:
vmax = √((Ncosθ * r) / m)
However, we need to determine the value of cosθ to get a precise a value for the maximum speed. To find the value of cosθ, we'll use the relationship between the radius and the height at the top of the hill.
Since the crest of the hill is a circular arc, the height (h) at the top of the hill is related to the radius and angle as follows:
h = r - r*cosθ
We can rearrange this equation to solve for cosθ:
cosθ = (r - h) / r
Substituting this value of cosθ into the equation for vmax:
vmax = √(((N * (r - h)) / r) / m)
To find the maximum speed, we need to determine the value of N. At the top of the hill, the forces acting on the motorcycle are the weight (mg) and the normal force (N), so the sum of these forces must be zero:
N - mg = 0
N = mg
Substituting this value of N into the equation for vmax:
vmax = √(((mg * (r - h)) / r) / m)
Now, we have an equation to find the maximum speed (vmax) in terms of the given variables (m, g, r, and h). Simply substitute the values of the mass of the motorcycle (m) and the acceleration due to gravity (g), and the given radius (r) and height (h) to calculate the maximum speed.