A child goes down a playground slide with an acceleration of 1.36 m/s^2.

1-Find the coefficient of kinetic friction between the child and the slide if the slide is inclined at an angle of 28.0 deg below the horizontal.

Oh, I see the child is enjoying a thrilling slide ride! Let's find that coefficient of kinetic friction, shall we?

First, we need to determine the force causing the acceleration. We can break this force into two components: the force parallel to the incline (causing acceleration) and the force perpendicular to the incline (due to the child's weight). We will focus on the parallel component.

Using trigonometry, we find that the force parallel to the incline can be represented by F_parallel = m * g * sin(theta), where m is the mass of the child, g is the acceleration due to gravity, and theta is the angle of the slide.

Now, let's relate the force parallel to the incline to the frictional force. The frictional force can be expressed as F_friction = mu_k * N, where mu_k is the coefficient of kinetic friction and N is the normal force (perpendicular force due to weight).

Since the force parallel to the incline is equal to the frictional force, we can equate these two expressions:

m * g * sin(theta) = mu_k * N

But what is the normal force N? It is equal to the weight of the child, which is m * g, multiplied by cos(theta):

N = m * g * cos(theta)

Now we can substitute N into our equation:

m * g * sin(theta) = mu_k * (m * g * cos(theta))

Canceling out m and g, we get:

sin(theta) = mu_k * cos(theta)

Finally, rearranging the equation to solve for the coefficient of kinetic friction:

mu_k = tan(theta)

Substituting the given angle of 28.0 degrees:

mu_k = tan(28.0)

Now it's time for you to grab your calculator and find the value of this tangent. Good luck, and remember to slide responsibly!

To find the coefficient of kinetic friction between the child and the slide, we need to use the equation that relates the acceleration to the forces involved.

The child's acceleration down the slide is given as 1.36 m/s^2. Let's assume the child's mass is m. The forces acting on the child are the force due to gravity (mg) and the force of kinetic friction (f_k).

To break the forces down into components, we can consider the inclined plane as having two perpendicular axes: one parallel to the incline (x-axis) and one perpendicular to the incline (y-axis). The force due to gravity can be divided into two components: the component along the x-axis (mg*sinθ) and the component along the y-axis (mg*cosθ).

Since the child is moving down the slide, the force of kinetic friction acts up the incline (opposite to the child's motion). Therefore, it can be expressed as a negative force (-f_k).

Using Newton's second law, we can write the equations of motion along the axes:

Along the x-axis: ma_x = -f_k
=> m(1.36 m/s^2) = -f_k

Along the y-axis: ma_y = mg*cosθ
=> m(0) = mg*cosθ

We can solve the second equation to find the value of cosθ:

mg*cosθ = 0
=> cosθ = 0

Since cosθ = 0, it means that the incline is vertical (90 degrees). This doesn't make sense for a slide that is inclined at an angle of 28.0 deg below the horizontal. Therefore, it seems there is an error in the problem statement or the given information.

Please ensure that the values and the question are correct, or provide any additional information if available.

To find the coefficient of kinetic friction between the child and the slide, we need to use the information given and some basic principles of physics.

First, let's identify all the known quantities:
- Acceleration (a) = 1.36 m/s^2 (given)
- Angle of inclination (θ) = 28.0 degrees (given)

Now, let's break down the forces acting on the child as it goes down the slide. There are two main forces at play: the force of gravity (mg) and the force of friction (f).

The force of gravity can be broken down into two components: the component perpendicular to the slide (mgcosθ) and the component parallel to the slide (mgsinθ). The force of friction acts in the opposite direction to the motion (down the slide).

Using Newton's second law of motion, we can write the equation of motion for the child:
mgsinθ - f = ma ...(1)

Now, we need to express the force of friction (f) in terms of the normal force (N) and the coefficient of kinetic friction (μ). The normal force (N) is the component of the force of gravity perpendicular to the slide, which can be calculated as N = mgcosθ.

The force of friction is given by the equation:
f = μN

Substituting the value of N in terms of mgcosθ and rearranging the equation, we get:
f = μmgcosθ

Now, we can substitute the value of f in equation (1):
mgsinθ - μmgcosθ = ma

Canceling out the mass (m) from both sides, we get:
gsinθ - μgcosθ = a

Finally, substituting the known values of g (acceleration due to gravity ≈ 9.8 m/s^2) and θ (28.0 degrees) into the equation, we can solve for the coefficient of kinetic friction (μ).

μ(9.8 * sin(28.0)) = 1.36

μ * 4.694 = 1.36

μ ≈ 0.29

Therefore, the coefficient of kinetic friction between the child and the slide is approximately 0.29.

force gravity down= mg sin28

friction down: mg*mu*cos28

netforce=mass*acceleration
mg(sin28-mu*cos28)=ma

solve for mu.