A child goes down a playground slide with an acceleration of 1.24 m/s^2.
Find the coefficient of kinetic friction between the child and the slide if the slide is inclined at an angle of 32.0 degrees below the horizontal.
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The childs weight has two components, down the slide, normal to the slide.
down the slide=mg*sinTheta
normal to slide=mg*cosTheta
so
net force = m*a
mg*sinTheta-mu*mg*cosTheta= ma
solve for mu.
To find the coefficient of kinetic friction between the child and the slide, we need to use Newton's second law and the concept of inclined planes.
Let's break down the problem step by step:
1. Calculate the net force acting on the child:
The net force can be calculated using the equation F_net = m * a, where F_net is the net force, m is the mass of the child, and a is the acceleration. In this case, the acceleration is given as 1.24 m/s^2.
2. Determine the weight of the child:
The weight of an object is given by the equation W = m * g, where W is the weight, m is the mass, and g is the acceleration due to gravity. We'll use g = 9.8 m/s^2 for simplicity.
3. Determine the normal force acting on the child:
The normal force is the force exerted by a surface to support the weight of an object resting on it. On an inclined plane, the normal force can be calculated as N = mg * cos(theta), where N is the normal force, mg is the weight of the child, and theta is the angle of inclination.
4. Calculate the force due to friction:
The force due to friction can be calculated using the equation F_friction = mu_k * N, where F_friction is the force due to friction and mu_k is the coefficient of kinetic friction.
5. Set up and solve the equation:
Since the child is moving down the slide, the net force is equal to the force due to friction:
F_net = F_friction
m * a = mu_k * N
m * a = mu_k * mg * cos(theta)
6. Substitute the given values:
The acceleration, a, is given as 1.24 m/s^2.
The angle of inclination, theta, is given as 32.0 degrees.
The mass of the child, m, is not provided.
We can rewrite the equation as:
m * 1.24 = mu_k * m * 9.8 * cos(32.0)
7. Solve for the coefficient of kinetic friction:
The mass, m, cancels out from both sides of the equation.
mu_k = (1.24) / (9.8 * cos(32.0))
Now you can calculate the coefficient of kinetic friction using the equation from step 7. Plug in the values and perform the calculations to find the answer.
To find the coefficient of kinetic friction, we can use the following steps:
Step 1: Draw a free-body diagram of the child on the slide to identify the forces involved.
In this case, the forces acting on the child are:
- The normal force (N), perpendicular to the slide.
- The weight of the child (mg), acting downward.
- The friction force (f), parallel to the slide.
Step 2: Break down the weight force into its components.
The weight force (mg) can be broken down into two components:
- The perpendicular component (mgcosθ), which is equal to the normal force (N).
- The parallel component (mgsinθ), which acts parallel to the slide.
Step 3: Write the equations of motion along the slide's direction.
Since the child is accelerating down the slide, we can write the following equation of motion:
f - mgsinθ = ma
Step 4: Calculate the normal force.
The normal force (N) is equal to the perpendicular component of the weight:
N = mgcosθ
Step 5: Substitute the value of N into the equation of motion.
f - mgsinθ = ma
f - mg * sinθ = ma
Step 6: Rearrange the equation to solve for the friction force (f).
f = ma + mg * sinθ
Step 7: Calculate the acceleration (a).
Given that the acceleration is 1.24 m/s^2, we can substitute this value into the equation to find the friction force (f).
f = (1.24)(m) + (m)(g)(sinθ)
Step 8: Solve for the coefficient of kinetic friction.
The coefficient of kinetic friction (μ) can be found using the equation:
μ = f/N
Substitute the value of f and N from the previous steps:
μ = [(1.24)(m) + (m)(g)(sinθ)] / (m)(g)(cosθ)
Simplify the equation:
μ = 1.24 + tanθ
Step 9: Substitute the angle θ and calculate the coefficient of kinetic friction.
Given that the slide is inclined at an angle θ = 32.0 degrees, we can substitute this value into the equation:
μ = 1.24 + tan(32.0)
Using a scientific calculator, calculate the value of μ:
μ ≈ 1.24 + 0.625
μ ≈ 1.865
Therefore, the coefficient of kinetic friction between the child and the slide is approximately 1.865.