10 - centre of mass 2

Question 1

CoM of a rod of length a with a variable denisty f(x) =

Answer 1

x = ∫xf(x)/∫f(x) from a-0

Question 1

finding CoM

Answer 1

when a rod has variable density f(x)
or a lamina has a shape f(x)
or a solid of revolution defined by f(x)
then you can integrate to find the centre of mass

Question 2

the centre of mass of a uniform lamina defined by f(x) from 0-a

Answer 2

x = ∫xf(x) /∫f(x)
y = 1/2 ∫f(x)² /∫f(x)
from a- 0

Question 3

solid of revolution

Answer 3

solid formed by rotating a function about an axis
CoM is always on the axis of revolution

Question 4

CoM of solid of revolution

Answer 4

x = ∫𝞹xy² / ∫𝞹y²

Question 5

equilibrium of a rigid body

Answer 5

if in equilibrium resuktant force and moment is 0

Question 6

suspending a lamina about a point

Answer 6

the line of action of W and reaction will pass through the point of suspension
CoM hang svertically below point of suspension

Question 7

toppling / sliding

Answer 7

if a lamina is on a rough plane it will slide or topple
- to slide the force needs to be greater than friction
- to topple the vertical line from the CoM must be outside its base