- #1
Mooshk
- 5
- 0
Homework Statement
First Problem:
A cyclist coasts down a 3 degree hill at a constant speed of 2.2 m/s. When the cyclist pumps hard, they can descend the hill at 12.6 m/s. Using the same power, what is the speed the cyclist can climb uphill? The cyclist and bicycle weigh 70 kg. Assume that frictional force = bv.
Second Problem:
A 0.2 kg box with a spring attached horizontally slides on a table where the coefficient of friction is 0.30. A force of 21 N can compress the spring 17cm. If the spring is released from this position, how far beyond its equilibrium will it stretch on the first swing?
Homework Equations
Force(friction) = bv
Power = W/t; Power = Force(v)
F(spring) = kx; k = F/x
Gravitational Potential Energy = mgh
Kinetic Energy = 1/2 mv^2
Elastic Potential Energy = 1/2 kx^2
Work(Friction) = change in total energy
(Based on my guesses at solving these problems)
The Attempt at a Solution
For the first problem:
mgsin3 / v = b
(70)(9.8)sin3 / 2.2 = b
16.32 = b
mgsin3 - bv + F(cyclist) = 0
35.90 - 205.63 + F = 0
169.73 N = F
P = F(cyclist) * v
P = 2138.60 W
Then, for going uphill,
-mgsin3 - bv + F = 0 ? Then I would solve for F again, and sub that into P=FV to find speed?
-----------
For the second problem:
I don't think I understand this correctly. If anyone can explain this problem to me, I'd appreciate it.
k = F/x
k = 21/0.17
k = 123.52
I think the initial and final kinetic energies are 0...
The initial potential elastic energy
E = 1/2 kx^2
E = 1/2 (123.53)(0.17 ^2)
E = 1.785 J
Force (friction) = (uk)mg
Force = (0.3)(0.2)(9.8)
Force = 0.588 N
Work (friction) = F(friction) * distance (or the change in total energy... I think I looked at this wrong)
Work = 0.588 N * distance
So the final elastic energy is
E = 1/2 k(distance ^2)
E = 1/2 (123.53) (distance ^ 2)
So then if I equate the two elastic energies:
1.785 J = 1/2 (123.53) (distance ^ 2)
3.57 = 123.53 (distance ^2)
0.0289 = (distance ^2)
0.17 m = distance
Now I'm not sure if I solved for the total distance traveled, or just past equilibrium.
I don't know if I understood these problems correctly. I'll appreciate any help you can give me. Thanks in advance.