A baby bounces up and down in her crib. Her mass is 13.5 kg, and the crib mattress can be modeled as a light spring with force constant 682 N/m.
If she were to use the mattress as a trampoline—losing contact with it for part of each cycle—what minimum amplitude of oscillation does she require?
Damon is Stating that-
F=1/T T=2*pi/w so T= 2pi/sqrt(k/m)
using our values T=2pi/sqrt(682/13.50)=.884004
Therefore F=1/.884004 -->=1.13121
For the second part acceleration=Aw^2 which must equal the acc. of gravity(g)
g=Aw^2 w=sqrt(k/m) so w=7.107638
plugging into our equation the equation g=Aw^2: 9.8=A*(7.107638^2)
9.8=50.518518*A
9.8/50.518518=A, so A=.193988 meters
which is about 19.39 cm
w = 2 pi f = sqrt(k/m)
x = A sin wt
v = A w cos wt
a = -A w^2 sin wt
max a = A w^2 = g
so
g = A w^2
A = g m/k
Well, if this baby wants to use the crib mattress as a trampoline, she better bring some spring to her step! With her mass of 13.5 kg and the force constant of the mattress being 682 N/m, we can use the equation for the minimum amplitude of oscillation (A) for a mass-spring system.
Now, the important thing to note is that the minimum amplitude occurs when the baby just loses contact with the mattress. When that happens, the force exerted by the mattress is just equal and opposite to the weight of the baby.
Using this information, the equation for the minimum amplitude of oscillation is given by:
A = (m * g) / k
where m represents the mass of the baby, g is the acceleration due to gravity (approximately 9.8 m/s²), and k is the force constant of the mattress.
Plugging in the values, we have:
A = (13.5 kg * 9.8 m/s²) / 682 N/m
Let's calculate that:
To find the minimum amplitude of oscillation required for the baby to lose contact with the mattress, we need to consider the forces acting on her.
The forces acting on the baby are the weight (mg) pulling her downward and the spring force (kx) pushing her upward. At the moment when the baby loses contact with the mattress, the spring force must be equal to or greater than the weight.
Let's break down the solution step by step:
1. Calculate the weight of the baby:
The weight (W) is given by the formula W = mg, where m is the mass of the baby and g is the acceleration due to gravity (approximately 9.8 m/s^2).
W = 13.5 kg * 9.8 m/s^2 = 132.3 N
2. Set up the equation for the force balance:
At the moment the baby loses contact with the mattress, the spring force (kx) must be equal to or greater than the weight (W). Therefore, we can write:
kx ≥ W
3. Substitute the values into the equation:
682 N/m * x ≥ 132.3 N
4. Solve for the minimum amplitude of oscillation (x):
Divide both sides of the equation by the force constant (682 N/m):
x ≥ 132.3 N / 682 N/m
x ≥ 0.194 m
Therefore, the minimum amplitude of oscillation required for the baby to lose contact with the mattress is approximately 0.194 meters.