A coin of 5.30 gram is placed 12.0 cm from the center of a turntable. The coin has static and kinetic coefficients of friction with the turntable surface of = 0.900 and = 0.470.

What is the maximum angular velocity with which the turntable can spin without the coin sliding?

The static friction coefficient, Us, determines when it starts sliding.

M*g*Us = M*R*w^2

Solve for w, the angular velocity.

w = sqrt(g*Us/R) = 8.57 rad/s

Well, the first question is whether the coin even knows how to slide. Maybe it wants to be a dancer, not a slider. But let's assume it's a slippery little coin who won't stay put.

To figure out the maximum angular velocity without the coin sliding, we can use the equation for static friction:

F_static = μ_static * N

Where F_static is the force of static friction, μ_static is the static coefficient of friction, and N is the normal force.

The normal force can be determined using the equation:

N = m * g

Where m is the mass of the coin and g is the acceleration due to gravity.

Now, let's plug in the values. The mass of the coin is given as 5.30 grams, which we'll convert to kilograms by dividing it by 1000:

m = 5.30 g / 1000 = 0.0053 kg

The acceleration due to gravity is approximately 9.8 m/s^2.

Now, let's calculate the normal force:

N = 0.0053 kg * 9.8 m/s^2 = 0.05194 N

Great, now we can go back to the static friction equation to find the maximum angular velocity:

F_static = μ_static * N

Solving for F_static:

F_static = μ_static * N

F_static = 0.900 * 0.05194 N

F_static = 0.046746 N

Now, let's talk about angular velocity. Angular velocity is given by the equation:

ω = √(μ_k * g * r)

Where ω is the angular velocity, μ_k is the kinetic coefficient of friction, g is the acceleration due to gravity, and r is the distance from the center of the turntable to the coin.

Plugging in the values:

ω = √(0.470 * 9.8 m/s^2 * 0.12 m)

ω = √(5.496 m^2/s^2)

ω ≈ 2.345 m/s

So, the maximum angular velocity with which the turntable can spin without the coin sliding is approximately 2.345 m/s. Now, let's hope the coin doesn't decide to quit its sliding gig and take up dance instead!

To determine the maximum angular velocity with which the turntable can spin without the coin sliding, we need to consider the balance of forces acting on the coin.

The force of static friction (fs) between the coin and the turntable surface can be calculated using the equation:

fs = µs * N

Where µs is the static coefficient of friction and N is the normal force acting on the coin.

The normal force (N) can be calculated using the equation:

N = m * g

Where m is the mass of the coin and g is the acceleration due to gravity.

Given that the mass of the coin is 5.30 grams (or 0.0053 kg), and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the normal force as follows:

N = 0.0053 kg * 9.8 m/s^2
N = 0.05194 N

Now, we can calculate the maximum static friction force using the static coefficient of friction of 0.900:

fs = 0.9 * 0.05194 N
fs = 0.046746 N

The maximum static friction force is 0.046746 N.

To prevent the coin from sliding, the force of static friction must provide the centripetal force required to keep the coin in circular motion. The centripetal force (Fc) can be calculated using the equation:

Fc = m * r * ω^2

Where m is the mass of the coin, r is the distance of the coin from the center of the turntable (12.0 cm or 0.12 m), and ω is the angular velocity of the turntable.

We can rearrange this equation to solve for ω:

ω^2 = Fc / (m * r)
ω = √(Fc / (m * r))

Substituting the values, we can calculate the maximum angular velocity (ω) as follows:

ω = √(0.046746 N / (0.0053 kg * 0.12 m))
ω = √(0.046746 / 0.000636)
ω ≈ √(73.385)
ω ≈ 8.57 rad/s

Therefore, the maximum angular velocity with which the turntable can spin without the coin sliding is approximately 8.57 rad/s.

To find the maximum angular velocity with which the turntable can spin without the coin sliding, we need to consider the forces acting on the coin.

1. First, let's find the normal force acting on the coin. The normal force is the force exerted by the turntable surface perpendicular to the coin.

The weight of the coin (mg) acts vertically downward, and the normal force (N) acts vertically upward. Since the coin is not moving vertically, the normal force must balance the weight of the coin.

Therefore, N = mg, where m is the mass of the coin and g is the acceleration due to gravity.

In this case, the mass (m) is given as 5.30 grams, which is equal to 0.0053 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s².

Substituting these values, we find N = 0.0053 kg × 9.8 m/s² = 0.05194 N.

2. Next, let's find the frictional force acting on the coin. The frictional force is given by the equation F_friction = μN, where μ is the coefficient of friction.

In this case, we have the static coefficient of friction (μ_static) and the kinetic coefficient of friction (μ_kinetic).

When the turntable is not moving, the frictional force must be static friction, given by F_friction_static = μ_staticN.

Substituting the values, F_friction_static = 0.900 × 0.05194 N = 0.04675 N.

3. Now, to find the maximum angular velocity, we need to consider the centripetal force acting on the coin. This force is provided by the frictional force between the coin and the turntable surface.

The centripetal force (F_centripetal) is given by the equation F_centripetal = mω²r, where m is the mass of the coin, ω is the angular velocity, and r is the distance from the center of the turntable to the coin.

We want to find the maximum angular velocity (ω_max) at which the coin does not slide. In this case, the frictional force (F_friction_static) provides the centripetal force (F_centripetal).

Substituting the values, F_friction_static = F_centripetal, so 0.04675 N = mω_max²r.

Since we know the mass of the coin (0.0053 kg) and the distance from the center of the turntable to the coin (12.0 cm = 0.12 m), we can rearrange the equation to solve for ω_max.

ω_max = √(F_friction_static / mr).

Substituting the values, ω_max = √(0.04675 N / (0.0053 kg × 0.12 m)).

Evaluating this expression, ω_max ≈ 7.244 rad/s.

Therefore, the maximum angular velocity with which the turntable can spin without the coin sliding is approximately 7.244 rad/s.