A tick is standing on a turntable at distance r from the centre. The static friction coefficient between the tick and table is us. The table is spinning at a constant angular frequency of w
a) what values of r is the bug most likely to slip at? small or large ?
b) The tick is walking along a radius and slips. What is the value of r when the tick slips?
omega = constant angular velocity
v = omega r
Ac = omega^2 r
slips when mu m g = m omega^2 r
or mu g = omega^2 r
so
the bigger r, the more likely the slip
r = mu g/omega^2
To determine the values of r at which the bug is most likely to slip, we need to consider the opposing forces acting on the tick. In this case, the two main forces are the static friction force and the centrifugal force.
a) The static friction force acts to prevent the tick from slipping on the turntable. It can be calculated using the equation F_static = us * N, where us is the static friction coefficient, and N is the normal force.
The normal force N is equal to the weight of the tick, which is given by N = mg, where m is the mass of the tick, and g is the acceleration due to gravity.
The centrifugal force is an outward force experienced by an object moving in a circular path. It can be calculated as F_centrifugal = m * r * w^2, where m is the mass of the tick, r is the distance of the tick from the center, and w is the angular frequency of the turntable.
To prevent the tick from slipping, the maximum static friction force must be greater than or equal to the centrifugal force. Therefore, we can equate the two forces:
us * N >= m * r * w^2
Since N = mg, we have:
us * mg >= m * r * w^2
The mass 'm' cancels out, and we are left with:
us * g >= r * w^2
From this equation, we can see that the tick is most likely to slip at larger values of r. As the distance from the center increases, the centrifugal force also increases, requiring a larger static friction force to prevent slipping.
b) If the tick is slipping while walking along a radius, then the static friction force is no longer strong enough to counteract the centrifugal force. In this case, we can set the static friction force equal to the maximum value it can achieve:
F_static = us * N = us * mg
Substituting the equation for N, we have:
us * mg = m * r * w^2
The mass 'm' cancels out again, and we can solve for the distance 'r':
us * g = r * w^2
Dividing both sides by w^2, we get:
r = (us * g) / w^2
This equation gives us the value of 'r' when the tick slips while walking along a radius.