A vertical spring stretches 13 cm when a 2.3 kg block is hung from its end. (a) Calculate the spring constant. This block is then displaced an additional 3.2 cm downward and released from rest. Find the (b) period, (c) frequency, (d) amplitude, and (e) maximum speed of the resulting SHM
k=2.3*9.8/.13 N/m
frequency = 1/2PI * sqrt(k/m)
period=1/f
amplitude=3.2cm
max speed:
1/2 m v^2=1/2 k (.032)^2 solve for v
a) Use hooke's law F=-kx
b) period=T=2PI*(m/k)^1/2
c) frequency=1/T=(k/m)^1/2*(1/2PI)
d) amplitude=x distance compressed/stretched (in this case 3.2 cm)
e) use conservation of energy to solve for speed: 1/2kx^2=1/2mv^2
To solve this problem, we will use Hooke's Law and the equation for simple harmonic motion (SHM).
(a) Calculating the spring constant:
According to Hooke's Law, the force exerted by a spring is proportional to the displacement. The formula for Hooke's Law is:
F = -kx
Where:
F is the force exerted by the spring (in Newtons),
k is the spring constant (in N/m),
x is the displacement from the equilibrium position (in meters).
In this case, we have the displacement (x) of 13 cm = 0.13 m and the force (F) due to the weight of the block, which can be calculated as:
F = mg
Where:
m is the mass of the block (in kg),
g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values, we have:
F = 2.3 kg * 9.8 m/s^2 = 22.54 N
Now, we can use the equation F = -kx to find the value of the spring constant (k). Rearranging the equation, we get:
k = -F / x
Substituting the known values, we have:
k = -22.54 N / 0.13 m ≈ -173.38 N/m
Note: The negative sign indicates that the spring force opposes the displacement. However, when discussing magnitudes, we can ignore the negative sign.
Therefore, the spring constant is approximately 173.38 N/m.
(b) Calculating the period:
The period (T) of an object undergoing simple harmonic motion (SHM) is the time it takes to complete one full oscillation. The period can be calculated using the formula:
T = 2π * √(m / k)
Where:
m is the mass of the block (in kg),
k is the spring constant (in N/m).
Substituting the given values, we have:
T = 2π * √(2.3 kg / 173.38 N/m) ≈ 2.45 seconds
Therefore, the period of the resulting SHM is approximately 2.45 seconds.
(c) Calculating the frequency:
The frequency (f) of an object undergoing SHM is the number of oscillations per second. It is the reciprocal of the period (T) and can be calculated as:
f = 1 / T
Substituting the value of T, we have:
f = 1 / 2.45 seconds ≈ 0.408 Hz
Therefore, the frequency of the resulting SHM is approximately 0.408 Hz.
(d) Calculating the amplitude:
The amplitude (A) of an object undergoing SHM is the maximum displacement from the equilibrium position. In this case, the block is displaced an additional 3.2 cm downward from the 13 cm (0.13 m) position.
Therefore, the amplitude of the resulting SHM is 0.13 m + 0.032 m = 0.162 m (or 16.2 cm).
(e) Calculating the maximum speed:
The maximum speed (v_max) of an object undergoing SHM occurs when it passes through the equilibrium position. At this point, all potential energy is converted to kinetic energy.
The maximum speed can be calculated using the equation:
v_max = A * 2π * f
Where:
A is the amplitude (in meters),
f is the frequency (in Hz).
Substituting the given values, we have:
v_max = 0.162 m * 2π * 0.408 Hz ≈ 0.421 m/s
Therefore, the maximum speed of the resulting SHM is approximately 0.421 m/s.