A 30-lb weight is attached to one end of a 6-ft uniform beam whose own weight is 20-lb. Where is the balance point of the system? I know the answer is "1.2 ft from the 30-lb weight" but I can't figure out how to get there. Please help. Thanks...

Let x be the distance (in feet) of the balance point from the end with the 30 lb weight.

The total moment about the balance point must be zero, if the beam is balanced. The beam's weight acts through the center of mass and exerts a moment of (3-x)*20 ft-lb about the pivot point.

This equals the moment (of opposite sign) due to the added weight, which is 30*x. The center of mass and the added weight are on opposite sides of the balance point.

Solve (3-x)*20 = 30x
60 - 20x = 30x
60 = 50x
x = ___

1.2

30x-20(6-x)=0

30x-120+20x=0
50x-120=0
50x=120
x=2.4 ft

the balance point is 2.4ft from the 30lb weight.

To find the balance point of the system, we need to consider the torques acting on the beam. The torque is the force multiplied by the distance from the fulcrum (or pivot point). In this case, the fulcrum is where the beam is balanced.

Let's denote the distance from the 30-lb weight to the fulcrum as x. Since the beam is 6 ft long, the distance from the 20-lb weight to the fulcrum is 6 - x.

The torque exerted by the 30-lb weight is its weight (30 lbs) multiplied by its distance (x ft) from the fulcrum, giving us 30x.

The torque exerted by the 20-lb weight is its weight (20 lbs) multiplied by its distance (6 - x ft) from the fulcrum, giving us 20(6 - x).

For the system to be in balance (no net torque), these two torques must be equal.

So, we can set up the equation:

30x = 20(6 - x)

Let's solve this equation step by step:

30x = 120 - 20x

Combine like terms:

30x + 20x = 120

50x = 120

Divide both sides by 50:

x = 120/50

x = 2.4 ft

Therefore, the balance point of the system is 2.4 ft from the 30-lb weight.

To find the distance from the 30-lb weight to the balance point, we subtract x from the total length of the beam:

6 ft - 2.4 ft = 3.6 ft

Hence, the balance point is 1.2 ft from the 30-lb weight.