Unit 4 - Problem 8 ==> Challenge with a spring

Given:

• The mass of an object is compressed into the spring for a distance of delta x.
• mass = 500 grams =  0.5 kilograms
• delta x = 10 cm = 0.1 meters
• The spring  has a spring constant of 500 newtons/meter
• There is a resistance section:
• Whose length is 0.8 meters
• And which exerts a force of 1 newton
• The mass moves up a curving ramp that is 0.5 meters high
• Two questions
• Where does the mass eventually come to rest on  the resistance section as measured from the spring?
• What is the maximum height above the ramp achieved by the mass?

Calculate: Initial energy stored by the mass compressing the spring

• $$ E_{initial} = \frac{1}{2} k {\Delta{x}}^2 $$ 
• $$ E_{initial} = 0.5 * 500 * 0.1^2 = 2.5\;{joules} $$

Calculate: Energy dissipated traveling over the friction

• $$ E_{friction} = F_{friction} L_{friction} $$
• $$ E_{friction} = 1 * 0.8 = 0.8\;{joules} $$

Calculate: Cycles over the friction to dissipate the initial energy

• A full cycle entails two passes over the friction.
• $$ E_{friction-cycle} = 2 E_{friction} = 2 * 0.8 = 1.6\;{joules}$$
• $$ cycles = \frac{E_{initial}}{E_{friction-cycle}} = \frac{2.5}{1.6} =  1.5625 $$
• 1 full cycle  occurs leaving 2.5 - 1.6 = 0.9 joules of energy left
• Another right hand crossing occurs leaving 0.9 - 0.8 = 0.1 joules to come back and stop in the resistance area 
• $$ \frac{0.1\;{joules}}{0.8\;{friction-joules}} = \frac{x_{stop-from-right\;{m}}}{{0.8\;{m}}}$$
• There the mass stops 0.1 meters into the resistance from the right
• Or 0.7 meters from the spring

Calculate: Maximum energy at take-off of ramp

• Obviously, the maximum energy will occur with the minimum crossings of the friction and this occurs on the first cycle.
• $$ E_{take-off} = E_{initial} - E_{friction} - E_{up-ramp} $$
• $$ E_{take-off} = 2.5 - 0.8 - (m*g*h) $$
• $$ E_{take-off} = 2.5 - 0.8 - (.5*10*.5) $$ 
• $$ E_{take-off} = 2.5 - 0.8 - 2.5 = -0.8\;{joules}$$ 

Calculate: Maximum height above ramp

• The mass never takes off nor ever gets to the top of the ramp.