Little Norma has mass 20kg and wants to use a 4.0-m board of mass 10kg as a seesaw. Her friends are busy, so Little Norma seesaws by herself by putting the support at the system's center of gravity when she sits on one end of the board. How far is she from the support point?
When they talk about the "system's" center of gravity, they mean the board and Norma together. The CG is x past the fulcrum (opposite Norma) and she sits 2-x meters from the fulcrum, with the fulcrum between her and the CG.
20*(2-x) = 10 x
40 - 20x = 10x
30x = 40
x = 4/3 meter
To determine how far Little Norma is from the support point on the board, we can use the principle of torque. The torque exerted by an object is the product of its mass, distance from the fulcrum (support), and the gravitational acceleration.
In this case, the total torque of the system should be zero, as it is in equilibrium. The torque exerted by Little Norma is given by the equation:
Torque (τ) = mass (m) * distance (d) * gravitational acceleration (g)
The gravitational acceleration is approximately 9.8 m/s^2.
Let's proceed to calculate the distance (d) using the given values:
Mass of Little Norma (m) = 20 kg
Mass of the board (M) = 10 kg
Distance from the support (d) = ?
Considering the torque on both sides of the fulcrum, we can write:
Torque exerted by Little Norma = Torque exerted by the board
(Mass of Little Norma) * (Distance from support) * (gravitational acceleration) = (Mass of the board) * (Distance from support) * (gravitational acceleration)
Now, we can solve for the distance (d):
20 kg * d * 9.8 m/s^2 = 10 kg * 2.0 m * 9.8 m/s^2
Simplifying the equation:
20d = 20
d = 1 meter
Therefore, Little Norma is 1 meter from the support point.
To solve this problem, we can use the principle of balance and the concept of center of gravity. The center of gravity is the point at which the entire weight of an object can be considered to act. In this case, we can assume that the mass of the board is concentrated at its center, and the mass of Little Norma is concentrated at her own center.
First, let's find the center of gravity of the entire system, which includes both Little Norma and the board:
Center of gravity of the system = (mass of Norma * position of Norma + mass of board * position of board) / (mass of Norma + mass of board)
Given:
Mass of Little Norma (m1) = 20 kg
Position of Little Norma (d1) = distance from the support point
Mass of the board (m2) = 10 kg
Position of the board (d2) = distance from the support point
Center of gravity of the system = (m1 * d1 + m2 * d2) / (m1 + m2)
Since the support point is at the center of gravity of the system, the system will be in balance.
In this case, the support point is at the center of the board. Since the board's mass is evenly distributed, its center is at its midpoint.
So, d2 = 4.0 m / 2 = 2.0 m
Now, substituting the given values into the formula:
Center of gravity of the system = (20 kg * d1 + 10 kg * 2.0 m) / (20 kg + 10 kg)
To maintain balance when sitting on one end of the board, the center of gravity of the system must be in the middle of the board. Therefore, the equation can be rewritten as:
d1 = (2.0 m * (20 kg + 10 kg) - 10 kg * 2.0 m) / 20 kg
Simplifying the equation:
d1 = (40 kg * 2.0 m) / 20 kg
d1 = 4.0 m
Therefore, Little Norma must sit 4.0 meters away from the support point to maintain balance on the seesaw.