a recent study of how mice negotiate turns, the mice ran around a circular 90 degree turn on a track with a radius of 0.15 m. The maximum speed measured for a mouse (mass = 18.5 g) running around this turn was 1.29 m/s. What is the minimum coefficient of friction between the track and the mouse’s feet that would allow a turn at this speed
To determine the minimum coefficient of friction between the track and the mouse's feet, we can use the concept of centripetal force.
The centripetal force required to keep an object moving in a circular path is given by the equation:
F = (m * v^2) / r
where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.
In this case, the mass of the mouse is given as 18.5 g, which is 0.0185 kg, the velocity is given as 1.29 m/s, and the radius of the circular path is given as 0.15 m. Plugging in these values into the equation, we get:
F = (0.0185 kg) * (1.29 m/s)^2 / 0.15 m
F = 0.247 kg * m/s^2
Now, since the centripetal force is provided by the frictional force between the mouse's feet and the track, we can find the minimum coefficient of friction (μ) using the equation:
F = μ * N
where N is the normal force acting on the mouse.
The normal force (N) can be calculated using the equation:
N = m * g
where g is the acceleration due to gravity. On Earth, the value of g is approximately 9.8 m/s^2.
Plugging in the values, we get:
N = (0.0185 kg) * (9.8 m/s^2)
N = 0.1813 kg * m/s^2
Now we can substitute the value of N back into the equation F = μ * N:
0.247 kg * m/s^2 = μ * 0.1813 kg * m/s^2
Simplifying, we find:
μ = 0.247 kg * m/s^2 / (0.1813 kg * m/s^2)
μ ≈ 1.36
Therefore, the minimum coefficient of friction between the track and the mouse's feet that would allow a turn at this speed is approximately 1.36.