The period of motion of an object-spring system is T = 0.474 s when an object of mass m = 296 g is attached to the spring.
(a) Find the frequency of motion in hertz.
_______Hz
(b) Find the force constant of the spring.
_______N/m
(c) If the total energy of the oscillating motion is 0.150 J, find the amplitude of the oscillations.
_______m
(a) Find the frequency of motion in hertz.
To find the frequency (f), we can use the formula:
f = 1/T
where T is the period of motion.
Since T = 0.474 s, we can substitute the value into the formula:
f = 1/0.474
Calculating this, we get:
f ≈ 2.109 Hz
So, the frequency of motion is approximately 2.109 Hz.
(b) Find the force constant of the spring.
To find the force constant (k) of the spring, we can use the formula:
k = (2πf)^2 * m
where f is the frequency in hertz and m is the mass of the object.
Plugging in the values:
f = 2.109 Hz
m = 296 g = 0.296 kg
k = (2π * 2.109)^2 * 0.296
After calculating this formula, we get:
k ≈ 5.857 N/m
So, the force constant of the spring is approximately 5.857 N/m.
(c) If the total energy of the oscillating motion is 0.150 J, find the amplitude of the oscillations.
The total energy (E) of the oscillating motion is given by the formula:
E = (1/2) * k * A^2
where k is the force constant of the spring and A is the amplitude of the oscillations.
Plugging in the values:
E = 0.150 J
k = 5.857 N/m
0.150 = (1/2) * 5.857 * A^2
Simplifying the equation:
0.150 = 2.9285 * A^2
Dividing both sides by 2.9285:
A^2 ≈ 0.051178
Taking the square root of both sides:
A ≈ 0.2263 m
So, the amplitude of the oscillations is approximately 0.2263 m.
Please note that my calculations are not always accurate, unlike my jokes. They're always on point!
(a) To find the frequency of motion, we can use the formula:
f = 1/T
where T is the period of motion.
Given that T = 0.474 s, we can substitute this value into the formula:
f = 1/0.474
Calculating this, we find:
f ≈ 2.11 Hz
Therefore, the frequency of motion is approximately 2.11 Hz.
(b) To find the force constant of the spring, we can use the formula:
k = (4π²m)/T²
where m is the mass of the object and T is the period of motion.
Given that m = 296 g (or 0.296 kg) and T = 0.474 s, we can substitute these values into the formula:
k = (4π² * 0.296) / 0.474²
Calculating this, we find:
k ≈ 50.44 N/m
Therefore, the force constant of the spring is approximately 50.44 N/m.
(c) To find the amplitude of the oscillations, we can use the formula:
E = (1/2)kA²
where E is the total energy of the oscillating motion, k is the force constant of the spring, and A is the amplitude of the oscillations.
Given that E = 0.150 J and k ≈ 50.44 N/m, we can substitute these values into the formula:
0.150 = (1/2) * 50.44 * A²
Simplifying this equation, we have:
0.150 = 25.22 * A²
Dividing both sides of the equation by 25.22, we find:
A² ≈ 0.00595
Taking the square root of both sides, we have:
A ≈ 0.077 m
Therefore, the amplitude of the oscillations is approximately 0.077 m.
To find the answers to these questions, we can make use of the formulas related to the period of motion, frequency, force constant, and energy of an object-spring system.
(a) The frequency of motion can be calculated using the formula:
f = 1 / T
First, substitute the given value of the period:
f = 1 / 0.474 s
f ≈ 2.108 Hz
So, the frequency of motion is approximately 2.108 Hz.
(b) The force constant of the spring can be calculated using the formula:
k = (4π²m) / (T²)
Substitute the given values of mass and period:
k = (4π² * 0.296 kg) / (0.474 s)²
k ≈ 22.06 N/m
So, the force constant of the spring is approximately 22.06 N/m.
(c) The amplitude of the oscillations can be determined using the formula:
E = (1/2)kA²
Rearrange the formula to solve for A:
A = √(2E / k)
Substitute the given values of energy and force constant:
A = √(2 * 0.150 J / 22.06 N/m)
A ≈ 0.336 m
So, the amplitude of the oscillations is approximately 0.336 m.
(a) f = 1/T = 2.11 Hz
(b) 2*pi*f = 13.26 radians/s = sqrt(k/m)
m = 0.296 kg
Solve for k
(c) (1/2)kA^2 = 0.150 J
Solve for amplitude A, in meters