A bicyclist traveling at 7 m/s rides around an unbanked curve. The coefficient of friction between the tires and the road is 0.38. What is the radius of the shortest turn that the bicyclist can safely make?
Can someone please respond on this one.
Fc = mvˆ2/radius
Friction = mg*(coefficient of friction)
So, mvˆ2/r = mg0.38
masses cancel out
vˆ2/r=g0.38
r= vˆ2/g0.38
r= 7ˆ2/9.81*0.38 = 13.144482 m
To find the radius of the curve, we can use the concept of centripetal force.
The centripetal force required to keep the bicyclist moving in a curved path is provided by the friction between the tires and the road. The formula for centripetal force is:
Fc = m * ac
Where Fc is the centripetal force, m is the mass of the object moving in a curved path, and ac is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
ac = v^2 / r
Where v is the velocity of the bicyclist and r is the radius of the curve.
Since the bicyclist is moving at a constant velocity of 7 m/s, the centripetal acceleration can be calculated as:
ac = (7 m/s)^2 / r
We also know that the frictional force is equal to the coefficient of friction multiplied by the normal force (Fn). The normal force in this case is equal to the weight of the bicyclist, which can be calculated as:
Fn = m * g
Where g is the acceleration due to gravity.
The frictional force (Ff) can then be calculated as:
Ff = µ * Fn
Substituting the value of Fn, the equation becomes:
Ff = µ * m * g
Since the frictional force (Ff) is equal to the centripetal force (Fc), we can equate the two equations:
µ * m * g = m * ac
Canceling out the mass (m) from both sides of the equation, we can solve for the radius (r):
µ * g = ac / r
Applying the value of the coefficient of friction (µ = 0.38) and the acceleration due to gravity (g = 9.8 m/s^2), we can solve for the radius (r):
0.38 * 9.8 m/s^2 = (7 m/s)^2 / r
0.37 * 9.8 m/s^2 = (49 m^2/s^2) / r
3.804 m/s^2 = 49 m^2/s^2 / r
Multiplying both sides of the equation by r:
3.804 m/s^2 * r = 49 m^2/s^2
Dividing both sides of the equation by 3.804 m/s^2:
r = 49 m^2/s^2 / 3.804 m/s^2
Simplifying the expression:
r ≈ 12.88 meters
Therefore, the radius of the shortest turn that the bicyclist can safely make is approximately 12.88 meters.