In a popular amusement park ride, a rotating cylinder of radius
R = 2.60 m
is set in rotation at an angular speed of 4.00 rad/s, as in the figure shown below. The floor then drops away, leaving the riders suspended against the wall in a vertical position. What minimum coefficient of friction between a rider's clothing and the wall is needed to keep the rider from slipping? Hint: Recall that the magnitude of the maximum force of static friction is equal to μsn, where n is the normal force—in this case, the force causing the centripetal acceleration.
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To find the minimum coefficient of friction between a rider's clothing and the wall, we first have to determine the magnitude of the normal force acting on the rider.
We can start by finding the centripetal acceleration of the rider. The centripetal acceleration can be calculated using the formula:
a = ω^2 * R
where ω is the angular speed and R is the radius of the rotating cylinder. Plugging in the given values:
a = (4.00 rad/s)^2 * 2.60 m
Next, we need to determine the normal force, which is the force causing the centripetal acceleration. In this case, it is the gravitational force acting on the rider, which can be calculated using the formula:
Fg = m * g
where m is the mass of the rider and g is the acceleration due to gravity. However, the mass cancels out when we consider the normal force, so we don't need to know the value of m.
Now, we can find the normal force by equating it to the gravitational force:
Fg = n
n = m * g
Since n is acting vertically upwards, the magnitude of the normal force is equal to the gravitational force.
Now, we can calculate the maximum static friction force by multiplying the magnitude of the normal force by the coefficient of friction:
fmax = μs * n
where μs is the coefficient of static friction.
To find the minimum coefficient of friction needed to keep the rider from slipping, we have to set the maximum static friction force equal to the force causing the centripetal acceleration:
fmax = m * a
Plugging in the relevant values and canceling out the mass:
μs * n = a
μs * (m * g) = (4.00 rad/s)^2 * 2.60 m
From this equation, we can solve for the minimum coefficient of static friction (μs) needed to keep the rider from slipping.