What is the coefficient of friction between a sled and a plane inclined 30 degrees from the horizontal if the sled just slides without accelerating when given an initial push?

To determine the coefficient of friction between a sled and a plane inclined at a certain angle, we can use the following steps:

Step 1: Identify the forces acting on the sled:
- The weight of the sled acting vertically downwards.
- The normal force exerted by the inclined plane perpendicular to it.
- The frictional force opposing the motion of the sled.

Step 2: Resolve the weight of the sled and the normal force into their respective components.
- The weight of the sled can be resolved into two components: one perpendicular to the inclined plane and one parallel to it.
- The normal force can also be resolved into two components: one perpendicular to the inclined plane and one parallel to it.

Step 3: Determine the equation for the forces that are parallel to the inclined plane.
- In this case, the only force acting parallel to the inclined plane is the component of the weight of the sled.
- The frictional force opposes the motion of the sled and acts parallel to the inclined plane.

Step 4: Set up the equation using Newton's second law.
- Since the sled is not accelerating, the net force acting on it is zero.
- The sum of the forces parallel to the inclined plane equals zero.

Step 5: Solve the equation to find the coefficient of friction.
- The equation for the forces parallel to the inclined plane is: μ * N - W_parallel = 0
- Rearrange the equation to solve for μ (the coefficient of friction).

Please provide the mass of the sled and the acceleration due to gravity, so we can continue with the calculation.

To find the coefficient of friction between a sled and a plane inclined 30 degrees from the horizontal, we can use the concept of static equilibrium. In this case, the sled is not accelerating, meaning that the force of friction acting on it balances the component of the gravitational force pulling it downwards along the inclined plane.

The force of gravity acting on the sled can be broken down into two components: one perpendicular to the plane (normal force) and one parallel to the plane (component of gravitational force). The component of gravitational force parallel to the plane is given by the equation:

F_parallel = m * g * sin(θ)

where m is the mass of the sled, g is the acceleration due to gravity, and θ is the angle of inclination.

The force of friction between the sled and the inclined plane is given by:

F_friction = μ * N

where μ is the coefficient of friction and N is the normal force acting on the sled.

Since the sled is not accelerating, the force of friction can be equated to the component of gravitational force acting downhill:

F_friction = F_parallel

Substituting the equations, we get:

μ * N = m * g * sin(θ)

To find the coefficient of friction (μ), we need to determine the normal force (N).

The normal force can be calculated by considering the vertical equilibrium of the sled. Along the inclined plane, the normal force balances the component of gravitational force perpendicular to the plane.

N - (m * g * cos(θ)) = 0

From this equation, we can solve for N:

N = m * g * cos(θ)

Now we can substitute the value of N into the earlier equation to solve for the coefficient of friction:

μ * (m * g * cos(θ)) = m * g * sin(θ)

Simplifying:

μ = (m * g * sin(θ)) / (m * g * cos(θ))

The mass (m) and the acceleration due to gravity (g) cancel out:

μ = sin(θ) / cos(θ)

Finally, we can simplify further using the trigonometric identity for tangent:

μ = tan(θ)

Therefore, in this case, the coefficient of friction (μ) between the sled and the inclined plane is equal to the tangent of the angle of inclination (θ). In other words, it is equal to tan(30 degrees).