A 17 kg girl slides down a playground slide that is 2.7 m high. When she reaches the bottom of the slide, her speed is 1.3 m/s. What is the coefficient of friction? Use energy theorem
Potential energy at start = m g h
Kinetic energy at finish = (1/2) m v^2
energy turned into heat = mgh - (1/2) m v^2
that is friction force times distance down slide
if angle of inclination of slide it T
distance down slide = h/sin T
normal force on slide = m g sin T
friction force = mu m g sin T
so energy to friction = mu m g sin T * h/sin T = mu m g h (remarkable)
so
mu m g h = m g h - (1/2) m v^2
note that m does not matter :)
To find the coefficient of friction, we can use the energy theorem.
The energy theorem states that the total mechanical energy of a system remains constant as long as no external forces are acting on it. In this case, we can consider the girl sliding down the slide as our system.
The total mechanical energy of the system can be calculated as the sum of the potential energy (PE) and the kinetic energy (KE), given by the following equations:
PE = mgh
KE = (1/2)mv^2
Here, m represents the mass of the girl (17 kg), g is the acceleration due to gravity (9.8 m/s^2), h is the height of the slide (2.7 m), and v is the velocity of the girl at the bottom of the slide (1.3 m/s).
Initially, when the girl is at the top of the slide, all her energy is in the form of potential energy, given by PE = mgh.
Final energy = KE = (1/2)mv^2
Since no external forces are acting on the girl-slide system, the initial mechanical energy (PE) is equal to the final mechanical energy (KE):
mgh = (1/2)mv^2
Now, we can solve this equation to find the coefficient of friction (μ). The coefficient of friction can be expressed as the ratio of the work done by friction to the initial potential energy of the girl:
μ = (Work done by friction) / (Initial potential energy)
It is important to note that the work done by friction is equal to the force of friction multiplied by the distance over which it acts. Since the girl is sliding down the slide, there is no horizontal displacement, and thus, no work is done by friction in the horizontal direction.
Therefore, the equation for μ becomes:
μ = ( Vertical work done by friction) / (Initial potential energy)
Since the vertical motion of the girl is purely dictated by the height of the slide, the vertical work done by friction is given by:
Vertical work done by friction = -μmgd
Here, d represents the distance over which frictional force acts vertically down the slide.
Substituting this into the equation for μ, we get:
μ = (-μmgd) / (mgh)
Now, we can substitute the known values into the equation and solve for μ:
μ = (-μ * 17 kg * 9.8 m/s^2 * d) / (17 kg * 9.8 m/s^2 * 2.7 m)
Simplifying the equation, we find:
μ = (-μd) / 2.7
Finally, we can multiply both sides of the equation by 2.7 and isolate the μ term:
μ + μd = 0
μ(1 + d) = 0
μ = 0
Therefore, the coefficient of friction in this scenario is zero, suggesting that there is no friction between the girl and the slide.