A motorbike starts from rest at time t=0 and begins to accelerate around a circular track as shown in the figure below. Eventually, at time t=t1 the motorbike reaches the maximum velocity possible without slipping off the track. What's the minimum length in meters the motorbike must travel between t=0 and t=t1?
Radius of track=10m
Details and assumptions
Assume that the friction coefficient everywhere on the race track is the same.
The motorbike wheels never slip on the track as it accelerates.
20
7.8
To determine the minimum length the motorbike must travel between t=0 and t=t1, we need to consider the relationship between velocity, acceleration, and distance.
Given that the motorbike starts from rest and accelerates around a circular track, we can use centripetal acceleration to calculate the maximum velocity it can reach without slipping off the track.
Centripetal acceleration is given by the formula:
ac = v^2 / r,
where ac is the centripetal acceleration, v is the velocity, and r is the radius of the circular track.
The friction force between the motorbike's tires and the track provides the necessary centripetal force to keep the bike moving in a circle. The maximum frictional force is given by the equation:
fmax = μN,
where fmax is the maximum frictional force, μ is the friction coefficient, and N is the normal force (equal to the weight of the motorbike).
At the maximum velocity without slipping, the frictional force reaches its maximum value and equals the necessary centripetal force:
fmax = m * ac,
where m is the mass of the motorbike.
Combining the equations, we have:
μN = m * v^2 / r.
Since we want to find the minimum length of the track, we can use the relation between velocity and distance:
v = s / t,
where v is the velocity, s is the distance traveled, and t is the time taken.
Substituting v in the previous equation, we have:
μN = m * (s / t)^2 / r.
Rearranging the equation to solve for distance s, we get:
s = sqrt(μ * N * r * t^2 / m)
Now, we can plug in the given values:
Radius of track, r = 10m
Friction coefficient, μ = constant
Mass of motorbike, m = constant
Time taken, t = t1 - 0 = t1
The only missing value is the normal force, N. The normal force is equal to the weight of the motorbike and can be calculated as:
N = m * g,
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Once you have the value of N, you can substitute it back into the equation and solve for s. This will give you the minimum length in meters the motorbike must travel between t=0 and t=t1.