A child with a mass of 83 kg is on top of a water slide that is 12 m. long and inclined 30o with the horizontal. How fast is the child going at the bottom of the slide is?
(a.) the slide is friction less and the
(b.) coefficient of friction is 0.30?
h/s=sinα
h=s•sinα
PE=KE
m•g•h = m•v²/2
v=sqrt(2gh)
(b)
PE=KE+W(fr)
m•g•h = m•v²/2 = +F(fr) •s
Solve for ‘v”
To solve this problem, we can use principles of physics, specifically those related to inclined planes and energy conservation.
First, let's calculate the height difference between the top and bottom of the slide.
The vertical height can be calculated using the formula:
height = length * sin(angle)
height = 12 m * sin(30°)
height = 12 m * 0.5
height = 6 m
(a) When the slide is frictionless, we can use the principle of conservation of energy, which states that the total mechanical energy is conserved. The initial potential energy at the top of the slide will be converted entirely into the kinetic energy at the bottom.
The potential energy (PE) at the top is given by:
PE = mass * gravity * height
PE = 83 kg * 9.8 m/s^2 * 6 m
PE = 4838.8 Joules (J)
At the bottom, all of this potential energy will be converted into kinetic energy (KE).
KE = 1/2 * mass * velocity^2
Rearranging the equation to solve for velocity:
velocity = sqrt(2 * KE / mass)
Substituting the values:
velocity = sqrt(2 * 4838.8 J / 83 kg)
velocity ≈ 10.4 m/s
Therefore, when the slide is frictionless, the child will be going approximately 10.4 m/s at the bottom.
(b) When the coefficient of friction is 0.30, there will be friction between the child and the slide. In this case, let's consider the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
The initial potential energy at the top of the slide will be converted into both kinetic energy and work done against friction.
The work done against friction is given by:
work = friction force * distance
Friction force = coefficient of friction * normal force
The normal force can be calculated using:
normal force = mass * gravity * cos(angle)
Substituting the values:
normal force = 83 kg * 9.8 m/s^2 * cos(30°)
normal force = 83 kg * 9.8 m/s^2 * 0.866
normal force ≈ 694.54 N
friction force = 0.30 * 694.54 N
friction force ≈ 208.36 N
work = 208.36 N * 12 m
work = 2500.32 J
From the work-energy principle:
work = KE + ΔPE
The change in potential energy is given by:
ΔPE = -PE (negative because the potential energy decreases)
ΔPE = -4838.8 J
Solving for kinetic energy:
KE = work + ΔPE
KE = 2500.32 J - 4838.8 J
KE ≈ -2338.48 J
As the kinetic energy cannot be negative, it means that it is entirely exhausted due to the friction. Therefore, the child will come to a stop at the bottom of the slide when there is a coefficient of friction of 0.30.
To solve this problem, we can use the principles of conservation of energy and the laws of motion.
(a) When the slide is frictionless, we can assume that the only forces acting on the child are gravity and the normal force exerted by the slide. The work-energy principle states that the work done on an object is equal to its change in kinetic energy.
1. Find the height difference between the top and bottom of the slide:
The slide is inclined at an angle of 30 degrees with the horizontal. We can use trigonometry to find the height difference (h) between the top and bottom of the slide.
h = 12 m * sin(30°)
= 6 m
2. Calculate the potential energy at the top of the slide:
The potential energy (PE) of an object is given by the formula:
PE = mass * gravitational acceleration * height
PE = 83 kg * 9.8 m/s^2 * 6 m
= 4863.6 J
3. Calculate the final velocity at the bottom of the slide:
Since the slide is frictionless, all the potential energy is converted to kinetic energy at the bottom of the slide.
Kinetic energy (KE) = 1/2 * mass * velocity^2
Setting KE equal to the calculated potential energy:
1/2 * 83 kg * velocity^2 = 4863.6 J
velocity^2 = (2 * 4863.6 J) / 83 kg
velocity = sqrt((2 * 4863.6 J) / 83 kg)
≈ 10.34 m/s
Therefore, when the slide is frictionless, the child will be going at approximately 10.34 m/s at the bottom of the slide.
(b) When the coefficient of friction is 0.30, we need to take into account the additional force due to friction. Friction opposes motion and can reduce the child's speed.
1. Calculate the force of friction:
The force of friction (F_friction) is given by the formula:
F_friction = coefficient of friction * normal force
The normal force (N) is equal to the mass of the child multiplied by the gravitational acceleration:
N = mass * gravitational acceleration
F_friction = 0.30 * (83 kg * 9.8 m/s^2)
= 243.06 N
2. Calculate the net force:
The net force (F_net) acting on the child is equal to the force due to gravity minus the force of friction:
F_net = mass * acceleration
F_net = mass * (gravitational acceleration - acceleration due to friction)
Rearranging the equation, we have:
acceleration due to friction = gravitational acceleration - (F_friction / mass)
acceleration due to friction = 9.8 m/s^2 - (243.06 N / 83 kg)
= 9.8 m/s^2 - 2.93 m/s^2
= 6.87 m/s^2
3. Calculate the final velocity at the bottom of the slide:
The acceleration (a) is given, and the initial velocity (u) is 0 since the child starts from rest. We can use the equation of motion to find the final velocity (v):
v^2 = u^2 + 2as
v^2 = 0 + 2 * (6.87 m/s^2) * 6 m
v = sqrt(2 * (6.87 m/s^2) * 6 m)
≈ 7.80 m/s
Therefore, when the coefficient of friction is 0.30, the child will be going at approximately 7.80 m/s at the bottom of the slide.