Homework SourseHow do the following properties change as length scale shrin
How do the following properties change as length scale shrinks down? Provide some justification for your answers.
Example:
Beam length Beam surface area Beam volume
L L2
L3
Mass (m)
Cantilever spring constant as all dimensions scale – bending
Cantilever spring constant as all dimensions scale – tension
Pressure exerted by a body on the ground due to gravity
Friction force (assume ? is constant – a simple, but poor assumption!)
Energy stored in a capacitor
RC time constant
Thermal time constant
| Example: | L L2 L3 |
| Mass (m) | |
| Cantilever spring constant as all dimensions scale – bending | |
| Cantilever spring constant as all dimensions scale – tension | |
| Pressure exerted by a body on the ground due to gravity | |
| Friction force (assume ? is constant – a simple, but poor assumption!) | |
| Energy stored in a capacitor | |
| RC time constant | |
| Thermal time constant |
Solution
Answer:
1) Pressure exerted by gravity on ground changes by L2 (it\'s gravitational force per unit area)
2) Friction force independent of lengh shrinkage L0 (As it is dimensions independent)
3) Energy stored in capacitor changes by L, ( since stored energy = 1/2 CV2 where stored energy depends on capacitance i.e two parallel plates C = epsilon A/d)
4) Time costant is indepedent of length shrinkage in RC R changes by L, and C changes by 1/L
5) cantilever spring constant k = ( E * w * t3) / (4 * L3)
it changes by L since it is amount of force per unit length
6) Cantilever spring constant as all dimensions scale – tension as L3
