A child sits on one side of a teeter totter of length 7.60 m, a distance of 49.7 cm away from the end of the beam. The child has a mass of 21.6 kg. Her mother has a mass of 43.6 kg. How far away from her end does she have to sit in order for the teeter tooter to be exactly balanced?
Is the center of the board located as the fulcrum. If so, then distance from center to child*child's mass = distance from center to mother*mother's mass. Note that the distances given in the problem are from the end and not the center.
To solve this problem, let's first understand the concept of torque, which is the force that tends to cause rotation around an axis.
Torque (τ) is calculated by multiplying the force (F) applied to a lever arm by the perpendicular distance (r) from the axis of rotation to the point of application of the force: τ = F × r.
In a balanced teeter-totter, the total torque on both sides of the axis of rotation is equal.
In this case, the child and the mother are located on opposite sides of the teeter-totter. Let's denote:
- Mass of the child = m_child = 21.6 kg
- Mass of the mother = m_mother = 43.6 kg
- Length of the teeter-totter = L = 7.60 m
- Distance of the child from the end = r_child = 49.7 cm = 0.497 m (since torque requires distances in meters)
As the teeter-totter is balanced, the torques on either side of the axis of rotation should be equal.
Hence, we can set up the following equation:
(m_child × g × r_child) = (m_mother × g × r_mother)
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
We can solve this equation to find the value of r_mother (distance from mother's end) that satisfies this condition.
We can rearrange the equation as follows:
r_mother = (m_child × g × r_child) / (m_mother × g)
Simplifying the equation:
r_mother = (21.6 kg × 9.8 m/s² × 0.497 m) / (43.6 kg × 9.8 m/s²)
Calculating:
r_mother = 0.113 m or 11.3 cm
Therefore, in order for the teeter-totter to be perfectly balanced, the mother needs to sit approximately 11.3 cm away from her end.