A 4.3-kg block slides down an inclined plane that makes an angle of 26° with the horizontal. Starting from rest, the block slides a distance of 2.1 m in 5.1 s. Find the coefficient of kinetic friction between the block and plane.

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To find the coefficient of kinetic friction between the block and the plane, we can use the following steps:

Step 1: Determine the net force acting on the block.
To do this, we need to find the gravitational force pulling the block down the incline. The gravitational force can be calculated using the formula Fgravity = m * g, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2). In this case, the mass of the block is given as 4.3 kg, so the gravitational force Fgravity = 4.3 kg * 9.8 m/s^2.

Step 2: Calculate the component of the gravitational force along the incline.
Since the incline makes an angle of 26° with the horizontal, we need to find the component of the gravitational force acting parallel to the incline. This can be done by multiplying the gravitational force by the cosine of the angle. So, the component of the gravitational force along the incline is Fparallel = Fgravity * cos(theta), where theta is the angle of the incline (26°).

Step 3: Determine the acceleration of the block.
We can use the distance traveled by the block and the time it takes to find the acceleration. The formula for distance traveled by an object with constant acceleration is d = (1/2) * a * t^2, where d is the distance, a is the acceleration, and t is the time. In this case, the block slides a distance of 2.1 m in 5.1 s, so we can solve the equation for acceleration: 2.1 m = (1/2) * a * (5.1 s)^2.

Step 4: Calculate the frictional force.
The frictional force can be determined using the equation Ffriction = µ * N, where µ is the coefficient of kinetic friction and N is the normal force. On an inclined plane, the normal force is given by N = m * g * cos(theta), where m is the mass and g is the acceleration due to gravity. So, the frictional force becomes Ffriction = µ * m * g * cos(theta).

Step 5: Equate the net force to the frictional force.
Since the block is sliding down the plane, the net force can be expressed as Fnet = Fparallel - Ffriction. Since the block is moving at a constant speed, the net force is equal to zero. Therefore, we can set the equation Fparallel - Ffriction = 0.

Step 6: Solve for the coefficient of kinetic friction.
Substituting the known values into the equation, we get Fgravity * cos(theta) - µ * m * g * cos(theta) = 0. We can rearrange this equation to solve for µ: µ * m * g * cos(theta) = Fgravity * cos(theta), then µ = (Fgravity * cos(theta)) / (m * g * cos(theta)). The cos(theta) cancels out, leaving us with µ = Fgravity / (m * g).

Step 7: Plug in the values and calculate.
Substituting the given values into the equation, we get µ = (4.3 kg * 9.8 m/s^2) / (4.3 kg * 9.8 m/s^2). The mass and acceleration due to gravity cancel out, resulting in a coefficient of kinetic friction µ = 1.

Therefore, the coefficient of kinetic friction between the block and the plane is 1.