3)
a) Calculate the unstretched length of a spring, which has a Hooke's constant of 40N/m and is 56cm long when supporting a stationary 800g object.
I did
mag = kx
(0.800)(9.81) = 40x
x = 0.20m
l-x = u
56-20 = u
u = 36cm
Is that right?
b) The period with which the object would oscillate vertically.
I did
T = 2pi sqrt(0.800/40)
and got 0.9s...is that right?
c) The amplitude that would be required so that the object's maximum speed while oscillating would be 160cm/s. Then suggest why an oscillation is not possible with this spring.
I don't understand how to calculate that.
6) If a flea has a mass of 2.0x10^2 micrograms jumps vertically to a height of 65mm and 75% of the energy comes from elastic potential energy stored in the protein, determine the initial quantity of elastic potential energy.
3(a) and (b) look OK. For 3(c), use
(1/2)MVmax^2 = (1/2) kX^2 and solve for the maximum amplitude X. If X exceeds the unstretched spring length, such an oscillation is not possible, since you can't compress the spring to a negative length.
For (6), use
(1/2)M g H = (0.75)E where E is the potential energy stored in protein and H is the height that the flea can jump.
I got 23cm using that formula for 3a. Should I add that to 36? I don't get it.
3a) To find the unstretched length of the spring, you can use Hooke's Law:
F = k * x
Where F is the force applied on the spring, k is the Hooke's constant, and x is the displacement from the unstretched position.
In this case, the force applied on the spring is the weight of the object, which can be calculated as:
F = m * g
Where m is the mass of the object and g is the acceleration due to gravity.
Plugging in the values, we have:
F = (0.800 kg) * (9.81 m/s^2) = 7.848 N
Now, we can rearrange Hooke's Law to solve for x:
x = F / k
Plugging in the values, we have:
x = 7.848 N / 40 N/m = 0.1962 m
To convert this to centimeters, we can multiply by 100:
x = 0.1962 m * 100 = 19.62 cm
So, the unstretched length of the spring is 19.62 cm.
3b) To find the period of oscillation, you can use the formula:
T = 2 * pi * sqrt(m / k)
Where T is the period, m is the mass of the object, and k is the Hooke's constant.
Plugging in the values, we have:
T = 2 * pi * sqrt(0.800 kg / 40 N/m) ≈ 2 * pi * 0.2828 s ≈ 1.777 s
So, the period with which the object would oscillate vertically is approximately 1.777 seconds.
3c) To find the amplitude that would be required for the object's maximum speed to be 160 cm/s, we can use the equation:
v_max = A * sqrt(k / m)
Where v_max is the maximum speed, A is the amplitude, k is the Hooke's constant, and m is the mass of the object.
Plugging in the values, we have:
160 cm/s = A * sqrt(40 N/m / 0.800 kg) ≈ A * 25.0 m/s
Solving for A, we have:
A = 160 cm/s / (25.0 m/s) ≈ 6.4 cm
So, an amplitude of approximately 6.4 cm would be required for the object's maximum speed to be 160 cm/s.
Regarding why an oscillation is not possible with this spring for the given conditions, it could be due to the fact that the maximum speed of the object (160 cm/s) exceeds the natural frequency of the system. When the maximum speed exceeds the system's natural frequency, the oscillation becomes unstable or impossible to sustain.