A 72- kg
skateboarder grinds down a hubba ledge that is 2.9 m
long and inclined at 19 ∘
below the horizontal. Kinetic friction dissipates half of her initial potential energy to thermal and sound energies.
Part A
Part complete
What is the coefficient of kinetic friction between her skateboard and the ledge surface?
To solve this problem, we can use the conservation of energy principle. The initial potential energy of the skateboarder is converted into kinetic energy as she slides down the hubba ledge, and then into thermal and sound energies due to the friction between the skateboard and the ledge surface. We can use the work-energy theorem to relate the work done by friction to the change in kinetic energy.
First, let's calculate the initial potential energy of the skateboarder:
U_i = mgh = (72 kg)(9.8 m/s^2)(2.9 m)(sin 19°) ≈ 238 J
where m is the mass of the skateboarder, g is the acceleration due to gravity, h is the height of the hubba ledge above the ground, and sin 19° is the vertical component of the length of the hubba ledge.
Half of this initial potential energy is dissipated by friction, so the final kinetic energy of the skateboarder is:
K_f = (1/2)mv^2 = (1/2)m[(2gh - U_i)/m]^2
where v is the final velocity of the skateboarder at the end of the hubba ledge, and 2gh is the total potential energy of the skateboarder at the top of the hubba ledge.
The work done by friction during the slide down the hubba ledge is:
W_friction = ΔK = K_f - 0 = K_f
where ΔK is the change in kinetic energy.
The work done by friction can also be expressed as:
W_friction = F_friction d = μ_k N d
where F_friction is the force of kinetic friction, d is the length of the hubba ledge, μ_k is the coefficient of kinetic friction between the skateboard and the ledge surface, and N is the normal force between the skateboard and the ledge surface. The normal force is equal to the weight of the skateboarder, which is:
N = mg
Putting it all together, we get:
K_f = μ_k N d
(1/2)m[(2gh - U_i)/m]^2 = μ_k mgd
Solving for μ_k, we get:
μ_k = [(1/2)(2gh - U_i)/d]/g
Plugging in the given values, we get:
μ_k ≈ 0.49
Therefore, the coefficient of kinetic friction between the skateboard and the ledge surface is approximately 0.49.