Track cycling events at the Olympics are conducted on a banked track that has a

Question

Track cycling events at the Olympics are conducted on a banked track that has a circular shape with a circumference of 250 m, and a bank angle of with respect to the horizontal. See this link for a picture. There is a coefficient of static friction µs at the tire/track interface and the bike/cyclist mass is 80.0 kg. For a coordinate system, take the +x direction as horizontal (towards the center of the track) and the +y direction as vertically upwards. Now consider a situation where the cyclist is moving around the track at a constant speed of 9.00 m/s (approximately 20 mph) and would slip down the banking if there was a small decrease in speed.

a) Calculate the net force on the cyclist (you must include the direction).

b) Write down equations expressing Newton’s second law in both the x and y directions.

c) Calculate both the frictional force and the normal force when = 45.0 degrees.

d) Determine µs. Recall 9.00 m/s is the minimum speed to not slip down the banking.

Explanation / Answer

given, circumference of the circular track, 2*pi*r = 250 m so r= 30.8089 m

bank angle = theta coefficient of static friciotn, k

mass of cyclist, m = 80 kg

speed, v = 9 m/s

decreasing the speed would make the cyclist slide down

a. Net force on the cyclist is the centripital force due to the normal reaction of the floor

F = mv^2/r = 80*9^2/30.8089 = 210.328 N

b. applying newtons second law in x and y directions

x direction

Nsin(theta) - fcos(theta)= m*a

y-direction

mg = Ncos(theta) + fsin(theta)

but f = k*N

so,

Nsin(theta) - kN = m*a

mg = Ncos(theta) + kNsin(theta)

c. theta = 45 deg

N(sin(45) - k) = 80*9^2/30.8089 = 210.328

and, 80*9.81 = N(1 + k)*sqroot(2)

(sin(45) - k) = (1 + k)210.328*sqroot(2)/80*9.81

sin(45) - k = (k + 1)*0.379

k = 0.237

Hnece normal force, N = 447.404 N

and friction force, f = kN = 106.034 N

d. k = 0.237 N

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.