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?
need more help
I suppose we assume that the beam is balanced around its center, 3.80 meters from each end.
Then figure out how far from the center the child is. Do everything in meters:
3.80 - .497 = 3.303 meters
Then how much moment (torque) does the child exert around the center?
Weight = 21.6 kg so 21.6 g Newtons times 3.303 meters = 71.34 g Newton meters
Now
How far must the mother sit from the center (remember they asked for fromthe end so we have to do that at the end of this) to balance the child?
call it x from the center
So the moment is
43.6 g x Newton meters
so
43.6 g x = 71.34 g
Notice that g, the acceleration of gravity, cancels out and we could have done this in kilograms but best to do it in force units, Newtons
x = 71.34/43.6 meters from the center
x = 1.64 meters from the center
which is 3.80 - 1.64 = 2.16 meters (or 216 cm) from her end
Thanks!
To find the position at which the mother should sit to balance the teeter-totter, we need to consider the torques acting on the system.
The torque (τ) acting on an object is given by the equation:
τ = r * F * sin(θ)
where:
- r is the distance from the pivot point to the object
- F is the force acting on the object
- θ is the angle between r and F
In this case, the torques acting on the teeter-totter are:
1) The torque exerted by the child:
τ_child = r_child * F_child * sin(θ_child)
2) The torque exerted by the mother:
τ_mother = r_mother * F_mother * sin(θ_mother)
Since the teeter-totter is balanced, the total torque acting on it is zero.
To balance the torques, we can set up an equation:
τ_child + τ_mother = 0
Substituting the values we have:
r_child * F_child * sin(θ_child) + r_mother * F_mother * sin(θ_mother) = 0
We know the length of the teeter-totter (7.60 m) and the distance of the child from the end (49.7 cm = 0.497 m). Let's assume the mother sits at a distance "x" from her end.
Now we can rewrite the equation:
0.497m * 21.6kg * g * sin(θ_child) + x * 43.6 kg * g * sin(θ_mother) = 0
where g is the acceleration due to gravity (9.8 m/s^2).
However, we need one more piece of information to proceed: the angle at which each person is sitting with respect to the horizontal. If we assume that both individuals are sitting exactly parallel to the ground, then sin(θ_child) = sin(θ_mother) = 0.
So the equation becomes:
0.497m * 21.6kg * g * 0 + x * 43.6 kg * g * 0 = 0
Since both terms are zero, we can conclude that the teeter-totter will be balanced regardless of the position of the mother. Therefore, the mother can sit anywhere on her side without affecting the balance of the teeter-totter.