a student places a 10 g coin on a disk at a point 0.02 m from the center of the disk. the student rotates the disk with a steadily increasing speed. at what speed will the coin start to slide off the disk if the coefficient of static friction between the coin and the disk is 0.47?
To determine the speed at which the coin will start to slide off the disk, we need to consider the forces acting on the coin and the conditions for the initiation of sliding.
The forces acting on the coin are the gravitational force (mg) acting downwards and the static friction force (fs) acting upwards. The critical point at which sliding starts occurs when the static friction force reaches its maximum value. Therefore, we need to find the maximum static friction force.
The maximum static friction force (fs_max) can be calculated using the equation:
fs_max = μs * N
where μs is the coefficient of static friction and N is the normal force. In this case, the normal force N is equal to the gravitational force acting on the coin, so N = mg.
Substituting the values into the equation, we get:
fs_max = μs * mg
Now, we need to determine the value of fs_max at the critical point where the coin is about to slide off the disk. This occurs when the static friction force reaches its maximum value, which is equal to fs_max.
At the critical point, the centripetal force acting on the coin is equal to the static friction force:
m * v^2 / r = fs_max
where m is the mass of the coin, v is the linear velocity of the coin, and r is the distance of the coin from the center of the disk.
Substituting the given values, we have:
(10 g) * (v^2 / 0.02 m) = (0.47) * (10 g) * (9.8 m/s^2)
Simplifying the equation, we can cancel out the mass (10 g) and solve for the velocity v:
v^2 = (0.47) * (9.8 m/s^2) * (0.02 m)
v^2 = 0.0938 m^2/s^2
v = √(0.0938 m^2/s^2)
Therefore, the speed at which the coin will start to slide off the disk is approximately 0.306 m/s.