A 76-kg skateboarder grinds down a hubba ledge that is 2.5 m long and inclined at 22 ∘ below the horizontal. Kinetic friction dissipates half of her initial potential energy to thermal and sound energies.
What is the coefficient of kinetic friction between her skateboard and the ledge surface?
Well, well, well, looks like we've got a skateboarding physicist here! How cool is that? Now, let's grind our way through this question, shall we?
First things first, let's calculate her initial potential energy. The formula for potential energy is mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
Given that the skateboarder's mass is 76 kg, we can substitute these values into the formula. Multiplying them together, we get her initial potential energy.
Next, we need to figure out how much energy is dissipated by kinetic friction. The question tells us that half of her initial potential energy is converted to thermal and sound energies. That means we have to divide her initial potential energy by 2.
Alright, let's put our detective hats on and find the coefficient of kinetic friction. The equation for the work done by friction is W = μN, where W is the work done, μ is the coefficient of kinetic friction, and N is the normal force.
The normal force is equal to the weight of the skateboarder, which is mg. The work done by friction, in this case, is the force of friction multiplied by the distance she grinds on the ledge.
We can use some trigonometry to find the distance she grinds. So, the distance is equal to the hypotenuse of a triangle with a base of 2.5 m (the length of the ledge) and an angle of 22 degrees (the inclination of the ledge).
Now that we know the distance, we can plug it into the work equation and solve for the coefficient of kinetic friction.
So, my friend, grab your calculator, crunch those numbers, and voilà! You've got yourself the coefficient of friction between her skateboard and the ledge surface.
But hey, remember, laughter is the best physics! So, keep grinding, keep laughing, and keep being awesome!
To find the coefficient of kinetic friction between the skateboard and the ledge surface, we'll use the conservation of energy principle.
1. The potential energy of the skateboarder at the top of the ledge is given by:
PE = m * g * h
where m is the mass of the skateboarder (76 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the ledge.
2. The initial potential energy is dissipated to thermal and sound energies, so the decrease in potential energy is equal to the sum of both energies:
PE_loss = Thermal_energy + Sound_energy
3. The decrease in potential energy is given by:
PE_loss = (1/2) * m * g * h
(since half of the initial potential energy is dissipated)
4. Now, we need to find the height of the ledge (h). We can consider the height of the ledge as the vertical component of the length of the ledge. To find h, we use:
h = L * sin(angle)
where L is the length of the ledge (2.5 m) and angle is the inclination angle (22 degrees).
h = 2.5 m * sin(22 ∘)
h ≈ 0.92 m
5. Substituting the values into the equation, we get:
(1/2) * m * g * h = Thermal_energy + Sound_energy
(1/2) * 76 kg * 9.8 m/s^2 * 0.92 m = Thermal_energy + Sound_energy
352.64 J = Thermal_energy + Sound_energy
6. Now, we need to convert the thermal and sound energies into their respective forms of kinetic energy. Since kinetic friction is dissipated as thermal energy, we can write:
Thermal_energy = friction_force * distance
where friction_force is the force due to kinetic friction and distance is the distance traveled along the ledge (2.5 m).
7. The work done due to kinetic friction can be calculated using:
Work = force * distance
where force is the force due to kinetic friction and distance is the distance traveled along the ledge (2.5 m).
8. The force due to kinetic friction can be calculated using the equation:
force = coefficient_of_friction * normal_force
where coefficient_of_friction is the coefficient of kinetic friction, and normal_force is the force perpendicular to the ledge surface.
9. The normal force can be calculated using:
normal_force = m * g * cos(angle)
where angle is the inclination angle (22 degrees).
10. Substituting the values into the equations, we get:
Work = force * distance
= (coefficient_of_friction * normal_force) * distance
Force = coefficient_of_friction * normal_force
= coefficient_of_friction * m * g * cos(angle)
11. Comparing with the given values, we have:
Work = Thermal_energy + Sound_energy
= 352.64 J
Force * distance = 352.64 J
12. Substitute the expressions for the force and distance:
(coefficient_of_friction * m * g * cos(angle)) * distance = 352.64 J
13. Solve for the coefficient_of_friction using the given values:
coefficient_of_friction = (Work / (m * g * cos(angle) * distance)).
Let's calculate the coefficient of kinetic friction using the given values.
To find the coefficient of kinetic friction between the skateboard and the ledge surface, we need to consider the kinetic energy and the work done by friction.
First, let's calculate the initial potential energy of the skateboarder. The formula for potential energy is given by:
Potential Energy = mgh
Where:
m = mass of the skateboarder (76 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height (using trigonometry, we can find h = 2.5 m * sin(22°), since the ledge is inclined at 22° below the horizontal)
Substituting the values into the formula, we have:
Potential Energy = (76 kg) * (9.8 m/s^2) * (2.5 m * sin(22°))
Next, we know that half of the initial potential energy is dissipated as thermal and sound energies due to friction. Therefore, the work done by friction can be calculated as:
Work Done by Friction = (1/2) * Potential Energy
Now, we need to relate the work done by friction with the force of friction and the distance over which it acts. The formula is given by:
Work Done by Friction = Force of Friction * Distance
Since the skateboarder travels the entire length of the ledge (2.5 m), the distance over which friction acts is equal to the length.
Substituting the values, we have:
(1/2) * Potential Energy = Force of Friction * (2.5 m)
Now, let's find the force of friction using the formula:
Force of Friction = Coefficient of Friction * Normal Force
Since the skateboarder is on a flat surface, the normal force is equal to the weight of the skateboarder which is given by:
Normal Force = mass * gravity
Substituting the values, we have:
(1/2) * Potential Energy = (Coefficient of Friction) * (mass * gravity) * (2.5 m)
Now, we can solve for the coefficient of kinetic friction:
Coefficient of Friction = [(1/2) * Potential Energy] / [(mass * gravity * distance)]
Substituting the values and calculating, the coefficient of kinetic friction can be found.