Max (15 kg) pushes maya (12 kg) on a swing so that maya moves in simple harmonic motion. the swing is 1.8 meters long and has a
2 answers:
The period of a simple pendulum depends only on the length of the pendulum and the gravitational acceleration:
where L is the pendulum length and g the gravitational acceleration.
The problem says that Maya and the swing form a simple pendulum, so we can use this formula to calculate the period of Maya's motion, using the length of the swing (L=1.8 m):
where L is the pendulum length and g the gravitational acceleration.
The problem says that Maya and the swing form a simple pendulum, so we can use this formula to calculate the period of Maya's motion, using the length of the swing (L=1.8 m):
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(a) 80 N/m
The spring constant can be found by using Hooke's law:
where
F is the force on the spring
k is the spring constant
x is the displacement of the spring relative to the equilibrium position
At the beginning, we have
F = 16.0 N is the force applied
x = 0.200 m is the displacement from the equilibrium position
Solving the formula for k, we find
(b) 0.84 Hz
The frequency of oscillation of the system is given by
where
k = 80 N/m is the spring constant
m = 2.90 kg is the mass attached to the spring
Substituting the numbers into the formula, we find
(c) 1.05 m/s
The maximum speed of a spring-mass system is given by
where
is the angular frequency
A is the amplitude of the motion
For this system, we have
(the amplitude corresponds to the maximum displacement, so it is equal to the initial displacement)
Substituting into the formula, we find the maximum speed:
(d) x = 0
The maximum speed in a simple harmonic motion occurs at the equilibrium position. In fact, the total mechanical energy of the system is equal to the sum of the elastic potential energy (U) and the kinetic energy (K):
where
k is the spring constant
x is the displacement
m is the mass
v is the speed
The mechanical energy E is constant: this means that when U increases, K decreases, and viceversa. Therefore, the maximum kinetic energy (and so the maximum speed) will occur when the elastic potential energy is minimum (zero), and this occurs when x=0.
(e) 5.51 m/s^2
In a simple harmonic motion, the maximum acceleration is given by
Using the numbers we calculated in part c):
we find immediately the maximum acceleration:
(f) At the position of maximum displacement:
According to Newton's second law, the acceleration is directly proportional to the force on the mass:
this means that the acceleration will be maximum when the force is maximum.
However, the force is given by Hooke's law:
so, the force is maximum when the displacement x is maximum: so, the maximum acceleration occurs at the position of maximum displacement.
(g) 1.60 J
The total mechanical energy of the system can be found by calculating the kinetic energy of the system at the equilibrium position, where x=0 and so the elastic potential energy U is zero. So we have
where
m = 2.90 kg is the mass
is the maximum speed
Solving for E, we find
(h) 0.99 m/s
When the position is equal to 1/3 of the maximum displacement, we have
so the elastic potential energy is
and since the total energy E = 1.60 J is conserved, the kinetic energy is
And from the relationship between kinetic energy and speed, we can find the speed of the system:
(i) 1.84 m/s^2
When the position is equal to 1/3 of the maximum displacement, we have
So the restoring force exerted by the spring on the mass is
And so, we can calculate the acceleration by using Newton's second law:
Answer:
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See attached file
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v = Velocity of sound
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Length of vocal tract is given by
At f = 270 Hz v = 750 m/s
At f = 2300 Hz v = 750 m/s
At f = 270 Hz v = 350 m/s
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