A child of mass 20.2 kg takes a ride on an irregularly curved slide of height 6.12 m. The child starts from rest at the top. The acceleration of gravity is 9.8 m/s2 .
If a frictional force acts on the child, what is the magnitude of the mechanical energy dissipated by this force, if the speed of the child at the bottom is 8.7 m/s?
force energy=initial PE-finalKE-final PE
= 20.2*9.8*6.12-1/2 20.2*8.7^2 - zero assuming the final height is zero.
Well, it seems like the frictional force is determined to rain on this child's parade! Let's calculate the magnitude of the mechanical energy dissipated by this force.
First, let's calculate the potential energy of the child at the top of the slide:
PE_top = mass * gravity * height
PE_top = 20.2 kg * 9.8 m/s^2 * 6.12 m
Next, we can calculate the kinetic energy of the child at the bottom of the slide:
KE_bottom = 0.5 * mass * velocity^2
KE_bottom = 0.5 * 20.2 kg * (8.7 m/s)^2
The difference between the potential energy at the top and the kinetic energy at the bottom represents the mechanical energy dissipated by the frictional force.
Mechanical energy dissipated = PE_top - KE_bottom
So, let's do the math:
Mechanical energy dissipated = (20.2 kg * 9.8 m/s^2 * 6.12 m) - (0.5 * 20.2 kg * (8.7 m/s)^2)
But let me tell you, calculating numbers isn't my strongest suit. I'd recommend plugging those numbers into a calculator to get the exact answer. Good luck!
To find the magnitude of the mechanical energy dissipated by the frictional force, we need to calculate the initial mechanical energy at the top of the slide and the final mechanical energy at the bottom of the slide. The difference between these two values will give us the magnitude of the mechanical energy dissipated.
1. Calculate the initial mechanical energy at the top:
The initial mechanical energy (E_initial) is given by the formula:
E_initial = m * g * h_top
where m is the mass of the child, g is the acceleration due to gravity, and h_top is the height of the slide.
Given:
m = 20.2 kg (mass of the child)
g = 9.8 m/s^2 (acceleration due to gravity)
h_top = 6.12 m (height of the slide)
Substituting these values into the formula, we get:
E_initial = 20.2 kg * 9.8 m/s^2 * 6.12 m
2. Calculate the final mechanical energy at the bottom:
The final mechanical energy (E_final) is given by the formula:
E_final = (1/2) * m * v_bottom^2
where v_bottom is the speed of the child at the bottom of the slide.
Given:
v_bottom = 8.7 m/s (speed of the child at the bottom)
Substituting these values into the formula, we get:
E_final = (1/2) * 20.2 kg * (8.7 m/s)^2
3. Calculate the mechanical energy dissipated:
The magnitude of the mechanical energy dissipated (E_dissipated) is given by the difference between the initial and final mechanical energies:
E_dissipated = E_initial - E_final
Substituting the values calculated in steps 1 and 2 into the formula, we get:
E_dissipated = (20.2 kg * 9.8 m/s^2 * 6.12 m) - ((1/2) * 20.2 kg * (8.7 m/s)^2)
Now, you can calculate the magnitude of the mechanical energy dissipated by performing the arithmetic calculations.
To find the magnitude of the mechanical energy dissipated by the frictional force, we need to first calculate the initial potential energy of the child at the top of the slide and then calculate the final kinetic energy of the child at the bottom of the slide. The difference between the two will give us the mechanical energy dissipated.
1. Calculate the initial potential energy (PE) at the top of the slide:
The potential energy formula is given by: PE = m * g * h
Where m is the mass of the child (20.2 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slide (6.12 m).
PE = 20.2 kg * 9.8 m/s^2 * 6.12 m = 1182.9848 J (Joules)
2. Calculate the final kinetic energy (KE) at the bottom of the slide:
The kinetic energy formula is given by: KE = 0.5 * m * v^2
Where m is the mass of the child (20.2 kg) and v is the velocity of the child at the bottom of the slide (8.7 m/s).
KE = 0.5 * 20.2 kg * (8.7 m/s)^2 = 776.871 J (Joules)
3. Calculate the mechanical energy dissipated:
Mechanical energy dissipated = Initial potential energy - Final kinetic energy
Mechanical energy dissipated = 1182.9848 J - 776.871 J = 406.1138 J (Joules)
Therefore, the magnitude of the mechanical energy dissipated by the frictional force is approximately 406.1138 Joules.