a child and a sled weighed 290N slide down an ice covered hill that is 10m high. at the bottom of the hill there is an free ice, rough horizontal surface that brings the sled to stop within a distance of 25m. what is the coefficient of fiction between the horizontal surface and the sled runners?
10 m * 290 N = 290 N * μ * 25 m
solve for μ (the coefficient of fiction)
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Uk=0.4
Well, it seems our little adventurous child and sled have gotten themselves into a slippery situation! Let's calculate the coefficient of friction to solve this chilly mystery.
We know the weight of the child and sled is 290N, which we can split into two components: the weight down the hill (mg sin θ) and the weight perpendicular to the hill (mg cos θ).
The weight down the hill is given by mg sin θ, where m is the mass of the child and sled, and θ is the angle of the hill. Since we know the height of the hill is 10m, we can use trigonometry to determine the angle θ.
With the height and the distance of the hill, we can use the relationship tan θ = height/distance to solve for θ. tan θ = 10/25, which means θ ≈ 0.38 radians.
Now, back to the weight down the hill, which is mg sin θ. Since the weight is given as 290N, we have 290 = m * 9.8 * sin 0.38. Solving for m, we find m ≈ 83.47 kg.
The weight perpendicular to the hill is mg cos θ, which is 83.47 kg * 9.8 m/s^2 * cos 0.38 = 808 N.
Now, let's focus on the stopping distance. The work done by friction is the force of friction multiplied by the stopping distance. The work done by friction is equal to the change in kinetic energy of the sled.
The initial kinetic energy of the sled is given by (1/2) * mass * velocity^2. Since the sled starts from rest, this initial kinetic energy is zero.
The final kinetic energy is also zero since the sled comes to a stop. Therefore, the work done by friction is zero.
The force of friction is equal to the weight perpendicular to the hill, which we determined to be 808 N.
To find the coefficient of friction (μ), we can use the equation for work (W) which is W = F * d * cos θ, where F is the force of friction, d is the stopping distance, and θ is the angle between the force and displacement vectors (which is 0 in this flat case).
Since work done by friction is zero, we have W = F * d * cos 0 = 0. But F is 808 N and d is 25 m, so we can solve for the coefficient of friction (μ): 0 = 808 * 25 * cos 0 * μ. Oops, we made a mistake!
It seems we forgot to convert the angle θ from radians to degrees in the equation above. So let's try again!
Using θ = 0.38 radians, we have W = F * d * cos(θ * (180/π)) = 0.
Substituting the values, we get 0 = 808 * 25 * cos(0.38 * (180/π)) * μ.
After evaluating the cosine expression, we find 0 = 808 * 25 * 0.927 * μ.
Finally, solving for μ, we get μ ≈ 0.92.
So, the coefficient of friction between the horizontal surface and the sled runners is approximately 0.92.
To find the coefficient of friction between the horizontal surface and the sled runners, we can start by calculating the gravitational potential energy of the child and sled at the top of the hill.
The gravitational potential energy (GPE) is given by the equation:
GPE = m * g * h
Where:
m = mass of the child and sled (unknown in this case)
g = acceleration due to gravity (approximated as 9.8 m/s^2)
h = height of the hill (10 m)
Since the child and sled have a weight of 290 N, we can convert the weight to mass using the equation:
Weight = m * g
Substituting the values:
290 N = m * 9.8 m/s^2
Now, we can calculate the mass:
m = 290 N / 9.8 m/s^2
m ≈ 29.59 kg
Next, we need to calculate the work done by friction to bring the sled to a stop. The work done by friction (W) is given by the equation:
W = force of friction * distance
In this case, the work done by friction is equal to the change in the gravitational potential energy. So, we can equate the two:
W = GPE = m * g * h
Since the sled comes to a stop, the work done by friction is negative and equal to the gravitational potential energy. Rearranging the equation, we get:
force of friction * distance = -m * g * h
Now we can substitute the given values:
force of friction * 25 m = -29.59 kg * 9.8 m/s^2 * 10 m
Simplifying the equation:
force of friction = (-29.59 kg * 9.8 m/s^2 * 10 m) / 25 m
force of friction ≈ -115.12 N
Since the force of friction is negative, we take the magnitude and it becomes positive. Therefore, the force of friction is approximately 115.12 N.
Finally, we can calculate the coefficient of friction (μ) using the equation:
force of friction = coefficient of friction * normal force
In this case, the normal force is equal to the weight of the sled, which is 290 N.
115.12 N = coefficient of friction * 290 N
Simplifying the equation:
coefficient of friction = 115.12 N / 290 N
coefficient of friction ≈ 0.397
Therefore, the coefficient of friction between the horizontal surface and the sled runners is approximately 0.397.