A small metal cylinder rests on a circular turntable that is rotating at a constant speed.

The small metal cylinder has a mass of 0.20kg, the coefficient of static friction between the cylinder and the turntable is 0.080, and the cylinder is located 0.15m from the center of the turntable.

Take the magnitude of the acceleration due to gravity to be 9.81 m/s^2.

What is the maximum speed v_max that the cylinder can move along its circular path without slipping off the turntable?

Oh, the wonders of friction! So, we have a metal cylinder enjoying a delightful spin on a turntable. To keep it from slipping off, we need to find its maximum speed, v_max.

First, we need to figure out the maximum static friction force, F_max, that can keep the cylinder in place. We can do this by multiplying the coefficient of static friction (μ_s) with the normal force (N).

Now, the normal force acting on the cylinder is the same as its weight. Let's calculate that:

N = m * g,
where m is the mass of the cylinder and g is the acceleration due to gravity.

Plugging in the given values:
N = 0.20 kg * 9.81 m/s^2.

Now that we have the normal force, we can find the maximum static friction force:

F_max = μ_s * N.

Plugging in the given coefficient of static friction:
F_max = 0.080 * (0.20 kg * 9.81 m/s^2).

Now, the maximum static friction force can be equated to the centripetal force that keeps the cylinder on its circular path. The centripetal force is given by the formula:

F_centripetal = m * v_max^2 / r,

where m is the mass of the cylinder, v_max is the maximum speed, and r is the radius at which the cylinder is located from the center of the turntable.

We can equate the two forces:

F_max = F_centripetal.

After a bit of rearrangement, we get:

v_max^2 = μ_s * g * r.

Plugging in the given values:
v_max^2 = 0.080 * (0.20 kg * 9.81 m/s^2) * 0.15 m.

Now, we just need to take the square root of both sides to find the maximum speed, v_max:

v_max = √((0.080 * (0.20 kg * 9.81 m/s^2) * 0.15 m)).

And there you have it! Calculate that expression to find the maximum speed, and your cylinder will be enjoying its circular path without slipping off the turntable.

To find the maximum speed v_max that the cylinder can move along its circular path without slipping off the turntable, we need to consider the balance of forces acting on the cylinder.

The only force that can potentially cause the cylinder to slip off the turntable is the static friction force acting inward towards the center of the turntable.

At the maximum speed v_max, the static friction force will be at its maximum value.

The static friction force can be calculated using the equation:

f_friction = μ_s * N

where μ_s is the coefficient of static friction and N is the normal force.

The normal force N can be calculated as the component of the weight of the cylinder acting perpendicular to the turntable.

N = mg * cos(θ)

where m is the mass of the cylinder, g is the acceleration due to gravity, and θ is the angle between the line connecting the cylinder's center to the center of the turntable and the vertical axis.

In this case, the angle θ is 90 degrees since the cylinder is located 0.15m from the center of the turntable.

Let's calculate the normal force N:

N = (0.20kg) * (9.81 m/s^2) * cos(90 degrees) = 0

Since cos(90 degrees) = 0, the normal force N is zero.

Therefore, the static friction force is also zero.

This means that there is no maximum speed v_max that the cylinder can move along its circular path without slipping off the turntable. The cylinder will slip off the turntable at any speed, as there is no static friction force to keep it in place.

To find the maximum speed that the cylinder can move along its circular path without slipping off the turntable, we need to consider the balance of forces acting on the cylinder.

The force of static friction between the cylinder and the turntable opposes the tendency for the cylinder to slide off due to its inertia. The maximum static friction force, Fs_max, can be calculated using the formula:

Fs_max = μs * N

where μs is the coefficient of static friction, and N is the normal force.

The normal force, N, can be calculated as the gravitational force acting on the cylinder:

N = mg

where m is the mass of the cylinder, and g is the acceleration due to gravity.

In this case, m = 0.20 kg, and g = 9.81 m/s^2.

Thus, N = (0.20 kg) * (9.81 m/s^2) = 1.962 N.

Now, we can calculate the maximum static friction force:

Fs_max = (0.080) * (1.962 N) = 0.157 N.

The maximum speed, v_max, can be calculated using the following equation:

Fs_max = m * a

where a is the centripetal acceleration of the cylinder.

The centripetal acceleration, a, can be calculated as:

a = (v^2) / r

where v is the velocity of the cylinder, and r is the radius of the circular path.

In this case, r = 0.15 m (given in the problem).

Thus, the equation for v_max becomes:

0.157 N = (0.20 kg) * [(v_max^2) / (0.15 m)]

Simplifying the equation, we can find the maximum speed, v_max:

v_max^2 = (0.157 N * 0.15 m) / (0.20 kg)

v_max^2 = 0.11775 Nm / kg

v_max = sqrt(0.11775 Nm / kg)

v_max ≈ 0.34 m/s

Therefore, the maximum speed v_max that the cylinder can move along its circular path without slipping off the turntable is approximately 0.34 m/s.