A 20 kg girl slides down a playground slide that's 3.2 m high. When she reaches the bottom of the slide her speed is 1.3 m/s. Assume any loss of energy is due to friction. If the slide is inclined at 20 degrees to the horizontal, what is the coefficient of kinetic friction between the fuel and the slide? (65.3N)
Well how much energy was lost to friction?
Potential at start = m g h = 20 * 3.2 * g = 64 g = 64*9.81 = 628 Joules
How much Kinetic energy was left at the bottom?
(1/2) m v^2 = .5 * 20 * 1.3^2 = 16.9 Joules
so
628 - 17 = 611 Joules went into heating up the slide.
That is the friction force * the slide distance
611 = Ff * 3.2/sin 20 = Ff * 3.2 /.342
so Ff = 65.3 Newtons (but that is not the answer. It asked for the coef.)
Ff = mu m g cos 20
65.3 = mu * 20 * 9.81 * .94
mu = 0.354
Well, isn't it a slippery situation we have here! Let's slide right into the problem, shall we?
Now, we know that the girl's mass is 20 kg, and she slides down a slide that's 3.2 m high at an inclined angle of 20 degrees to the horizontal. Her final speed at the bottom is 1.3 m/s, and we're told there's a force of 65.3 N of friction acting on her.
To tackle this, we'll start by finding the girl's potential energy (PE) at the top of the slide, which is equal to her mass (m) multiplied by the acceleration due to gravity (g) multiplied by the height of the slide (h):
PE = m * g * h
PE = 20 kg * 9.8 m/s^2 * 3.2 m
Now, let's find her final kinetic energy (KE) at the bottom of the slide. We can use the formula:
KE = (1/2) * m * v^2
KE = (1/2) * 20 kg * (1.3 m/s)^2
Next, we need to account for the work done by friction. The work done by friction is equal to the force of friction (F) multiplied by the distance over which it acts (d). In this case, the distance is the length of the slide (3.2 m). So:
Work = F * d
Now, since work is equal to the change in energy (ΔE), we can set up the following equation:
Work = ΔE = KE - PE
F * d = (1/2) * m * v^2 - m * g * h
Plugging in the given values and solving for the coefficient of kinetic friction (μ_k), we have:
65.3 N * 3.2 m = (1/2) * 20 kg * (1.3 m/s)^2 - 20 kg * 9.8 m/s^2 * 3.2 m
To find μ_k, we divide both sides of the equation by the weight of the girl (m * g):
μ_k = (1/20 kg * 9.8 m/s^2 * 20 kg * (1.3 m/s)^2 - 65.3 N * 3.2 m) / (20 kg * 9.8 m/s^2)
Simplifying this equation, we'll magically find the coefficient of kinetic friction!
And voila! We have our answer. Just make sure to plug in the values correctly, or things might get a bit slippery.
To find the coefficient of kinetic friction between the girl and the slide, we need to consider the forces acting on the girl as she slides down.
First, let's determine the gravitational potential energy of the girl at the top of the slide using the formula:
Potential Energy = mass * acceleration due to gravity * height
The mass of the girl is 20 kg, acceleration due to gravity is approximately 9.8 m/s^2, and the height of the slide is 3.2 m. Substituting these values, we get:
Potential Energy = 20 kg * 9.8 m/s^2 * 3.2 m
Potential Energy = 627.2 Joules
Next, let's find the kinetic energy of the girl at the bottom of the slide using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
The mass of the girl is 20 kg, and her velocity at the bottom of the slide is 1.3 m/s. Substituting these values, we get:
Kinetic Energy = (1/2) * 20 kg * (1.3 m/s)^2
Kinetic Energy = 16.9 Joules
Since energy is conserved, the potential energy at the top is equal to the kinetic energy at the bottom, so:
Potential Energy = Kinetic Energy
627.2 Joules = 16.9 Joules
Now, let's calculate the work done against friction. The work done is equal to the initial energy minus the final energy. In this case, the work done is equal to the initial potential energy minus the final kinetic energy:
Work done against friction = Potential Energy - Kinetic Energy
Work done against friction = 627.2 Joules - 16.9 Joules
Work done against friction = 610.3 Joules
The work done against friction is also equal to the product of the force of friction and the distance over which the force is applied. In this case, the distance traveled is equal to the height of the slide, which is 3.2 m.
So, the equation becomes:
Work done against friction = force of friction * distance
610.3 Joules = force of friction * 3.2 m
To find the force of friction, we can rearrange the equation:
force of friction = 610.3 Joules / 3.2 m
force of friction ≈ 190.7 N
Finally, to calculate the coefficient of kinetic friction, we can use the equation:
force of friction = coefficient of kinetic friction * normal force
From the given information, we know that the normal force is 65.3 N.
So, the equation becomes:
190.7 N = coefficient of kinetic friction * 65.3 N
To find the coefficient of kinetic friction, we can rearrange the equation:
coefficient of kinetic friction ≈ 190.7 N / 65.3 N
coefficient of kinetic friction ≈ 2.92
Therefore, the coefficient of kinetic friction between the girl and the slide is approximately 2.92.
To find the coefficient of kinetic friction between the girl and the slide, we can use the principles of energy conservation.
Firstly, we need to determine the initial potential energy and the final kinetic energy of the girl.
The initial potential energy (PE_initial) is given by:
PE_initial = m * g * h
where m is the mass of the girl (20 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slide (3.2 m).
PE_initial = 20 kg * 9.8 m/s^2 * 3.2 m
PE_initial = 627.2 J
The final kinetic energy (KE_final) of the girl is given by:
KE_final = (1/2) * m * v^2
where v is the final speed of the girl (1.3 m/s).
KE_final = (1/2) * 20 kg * (1.3 m/s)^2
KE_final = 16.9 J
The work done by the kinetic friction force (W_friction) can be calculated using the following equation:
W_friction = KE_final - PE_initial
W_friction = 16.9 J - 627.2 J
W_friction = -610.3 J
Since the work done by friction is negative, it indicates that there was a loss of energy due to friction.
The work done by friction can also be calculated as the product of the friction force (F_friction) and the distance traveled (d) along the slide:
W_friction = F_friction * d
The distance traveled along the slide (d) can be calculated using the height of the slide (h) and the inclination angle (θ) as follows:
d = h / sin(θ)
d = 3.2 m / sin(20°)
d = 9.47 m
Therefore, we have:
W_friction = F_friction * 9.47 m
We can rearrange the equation to solve for the friction force (F_friction):
F_friction = W_friction / 9.47 m
F_friction = -610.3 J / 9.47 m
F_friction = -64.5 N
The friction force can also be calculated as the product of the normal force (N) and the coefficient of kinetic friction (μ):
F_friction = μ * N
We are given the normal force (N) as 65.3 N.
Therefore, we can rearrange the equation to solve for the coefficient of kinetic friction (μ):
μ = F_friction / N
μ = -64.5 N / 65.3 N
μ ≈ -0.988
Since the coefficient of kinetic friction cannot be negative, we can take the absolute value to get:
μ ≈ 0.988
Therefore, the coefficient of kinetic friction between the girl and the slide is approximately 0.988.