A girl of 50 kg slides down a 8 m long slide into a pool. The upper end of the slide is 4.0 m above the water and the lower end of the slide is at the level of the water. Assume a coefficient of friction of 0.20, find: a) the potential energy of the girl at the start of her slide, b) the force due to friction, c) the energy loss in overcoming friction, d) maximum kinetic energy of the girl, e) the girls maximum velocity.
To answer the given questions, we need to use principles from classical mechanics, specifically related to potential energy, work, and kinetic energy. Here's a step-by-step explanation of how you can find the solutions:
a) The potential energy of the girl at the start of her slide can be calculated using the formula:
Potential energy (PE) = mass (m) × gravity (g) × height (h)
In this case, the mass is given as 50 kg, the gravity is approximately 9.8 m/s^2, and the height is 4.0 m. Substituting these values into the formula:
PE = 50 kg × 9.8 m/s^2 × 4.0 m
= 1960 Joules
Therefore, the potential energy of the girl at the start of her slide is 1960 Joules.
b) The force due to friction can be calculated using the formula:
Force of friction (Ff) = coefficient of friction (μ) × normal force (N)
The normal force in this case is the weight of the girl, which can be calculated using the formula:
Weight (N) = mass (m) × gravity (g)
Substituting the given mass value of 50 kg and gravity value of 9.8 m/s^2:
Weight (N) = 50 kg × 9.8 m/s^2
= 490 N
Now, substituting the coefficient of friction given as 0.20 and the calculated normal force:
Ff = 0.20 × 490 N
= 98 N
Therefore, the force due to friction is 98 Newtons.
c) The energy loss in overcoming friction is equal to the work done by the frictional force (Ff) over the distance (d) of the slide. The work done can be calculated using the formula:
Work (W) = Force (F) × distance (d) × cos(θ)
Here, the force is the force of friction (Ff) calculated in the previous step, the distance d = 8.0 m (length of the slide), and θ = 180 degrees (as the angle between the force and the displacement is 180 degrees).
W = -Ff × d × cos(θ)
= -98 N × 8.0 m × cos(180°)
= -98 N × 8.0 m × (-1)
Since the cosine of 180 degrees is -1, the work done by friction is:
W = 98 N × 8.0 m
= -784 Joules
The negative sign indicates that the energy is lost due to friction.
Therefore, the energy loss in overcoming friction is 784 Joules.
d) The maximum kinetic energy of the girl is equal to the difference between the potential energy at the start (PE) and the energy loss due to friction (W):
Kinetic energy (KE) = PE - |W|
= 1960 J - 784 J
= 1176 Joules
Therefore, the maximum kinetic energy of the girl is 1176 Joules.
e) The maximum velocity of the girl can be calculated using the formula for kinetic energy:
KE = (1/2) × mass × velocity^2
Rearranging the formula, we get:
Velocity (v) = √(2 × KE / mass)
Substituting the calculated kinetic energy of 1176 Joules and the mass of 50 kg:
v = √(2 × 1176 J / 50 kg)
= √(2352 J / 50 kg)
= √(47.04 m^2/s^2 / kg)
≈ √(47.04) m/s
≈ 6.86 m/s
Therefore, the girl's maximum velocity is approximately 6.86 m/s.
By following these steps, you can find the desired values related to the given scenario.