A 82 kg adult sits at one end of a 8.6 m long board. His 23 kg child sits on the other end. Where should the pivot be placed so that the board is balanced, ignoring the board's mass? Find the pivot point if the board is uniform and has a mass of 15 kg. Please help with step-by-step explanation for both parts! Thanks!

notice the mass of the board and loads are (massboard+82+15)

One can start anywhere, let the end with the adult be zero.

Sum moments about that end.

boardweight*4.3+23g*8.6-piviotlength*(massboard+loads)=0

lets solve it...

15*g*4.3+23*4.6-(massboard+82+23)g=0
the first part lets massboard be zero.
the second part lets it be 15kg.

In either case, solve for L, the position of the piviot point from the adult end.

To find the pivot point where the board is balanced, we need to consider the moments of the adult and the child relative to the pivot point. The moment is the product of the weight (mass × gravity) and the distance from the pivot point.

Part 1: Ignoring the board's mass

Step 1: Calculate the moments of the adult and the child relative to the pivot point.

Moment of adult = weight of adult × distance from pivot point
= 82 kg × 9.8 m/s² × (8.6 m - x)

Moment of child = weight of child × distance from pivot point
= 23 kg × 9.8 m/s² × x

Note: The distance from the pivot point to the adult is (8.6 m - x) since the pivot point will be at a distance x from the end where the adult is sitting, and the total length of the board is 8.6 m.

Step 2: Set the moments of the adult and the child equal to each other to achieve balance.

82 kg × 9.8 m/s² × (8.6 m - x) = 23 kg × 9.8 m/s² × x

Step 3: Solve the equation for x, which represents the pivot point.

739.76 m²/kg × (8.6 m - x) = 225.4 m²/kg × x

7397.6 m² - 739.76 m²/kg × x = 225.4 m²/kg × x

964.36 m²/kg × x = 7397.6 m²

x = 7397.6 m² / 964.36 m²/kg
x ≈ 7.66 m

Therefore, the pivot point should be placed approximately 7.66 meters from the end where the adult is sitting.

Part 2: Considering the board's mass

Step 1: Calculate the moments of the adult, the child, and the board relative to the pivot point.

Moment of adult = weight of adult × (distance from pivot point + half the length of the board)
= 82 kg × 9.8 m/s² × (8.6 m - x + 0.5 m)

Moment of child = weight of child × (distance from pivot point - half the length of the board)
= 23 kg × 9.8 m/s² × (x - 0.5 m)

Moment of board = weight of board × half the length of the board
= 15 kg × 9.8 m/s² × 0.5 m

Step 2: Set the moments of the adult, child, and board equal to each other to achieve balance.

82 kg × 9.8 m/s² × (8.6 m - x + 0.5 m) = 23 kg × 9.8 m/s² × (x - 0.5 m) + 15 kg × 9.8 m/s² × 0.5 m

Step 3: Solve the equation for x, which represents the pivot point.

758.16 m²/kg × (8.6 m - x + 0.5 m) = 225.4 m²/kg × (x - 0.5 m) + 73.5 m²

7581.6 m² - 758.16 m²/kg × x + 37.908 m²/kg = 225.4 m²/kg × x - 112.7 m²

982.628 m²/kg × x = 7443.708 m²

x = 7443.708 m² / 982.628 m²/kg
x ≈ 7.57 m

Therefore, when considering the board's mass, the pivot point should be placed approximately 7.57 meters from the end where the adult is sitting.

Kpmwpmwpu :-kgnpg

To many morons who cant explain...