An object of an unknown mass is attached to an ideal spring with force constant 120 N/m and is found to vibrate with a frequency of 6.00 Hz. Find (a) the period, (b) the angular frequency, and (c) the mass of this object.

T= 1/Frequency

angular freq= 2PI f

finally mass from this equation

Period=2PI sqrt(Mass/k)

8.22

To solve this problem, we can use the equations related to simple harmonic motion and the properties of a mass-spring system.

(a) The period (T) of an object undergoing simple harmonic motion can be found using the formula:

T = 1 / f

where f is the frequency.

Plugging in the given frequency of 6.00 Hz, we have:

T = 1 / 6.00 = 0.167 s

So, the period is 0.167 seconds.

(b) The angular frequency (ω) of an object undergoing simple harmonic motion can be found using the formula:

ω = 2πf

where f is the frequency.

Plugging in the given frequency of 6.00 Hz, we have:

ω = 2π * 6.00 = 37.7 rad/s

So, the angular frequency is 37.7 radians per second.

(c) The mass (m) of the object can be found using the formula:

ω = √(k / m)

where k is the force constant (spring constant) and m is the mass of the object.

Rearranging the formula, we have:

m = k / ω^2

Plugging in the given force constant of 120 N/m and the angular frequency of 37.7 rad/s, we have:

m = 120 / (37.7)^2 = 0.0881 kg

So, the mass of the object is 0.0881 kilograms.

To find the answers to these questions, we need to use the formulas for the period, angular frequency, and mass of an object attached to a spring.

(a) The period (T) is the time it takes for one complete vibration or oscillation. It is the reciprocal of the frequency (f). The formula to find the period is:

T = 1 / f

In this case, the frequency (f) is given as 6.00 Hz. Therefore, we can calculate the period as:

T = 1 / 6.00 Hz = 0.1667 s

So, the period is 0.1667 seconds.

(b) The angular frequency (ω) is a measure of how quickly the object oscillates back and forth. It is related to the frequency by the equation:

ω = 2πf

where π is a mathematical constant approximately equal to 3.14159. Substituting the given frequency of 6.00 Hz, we can calculate the angular frequency as:

ω = 2π × 6.00 Hz = 37.68 rad/s

So, the angular frequency is 37.68 radians per second.

(c) The mass (m) of the object can be determined using the formula for the angular frequency of a mass-spring system:

ω = √(k / m)

where k is the spring constant. Rearranging the formula, we can solve for the mass (m) as:

m = k / ω^2

In this case, the spring constant (k) is given as 120 N/m, and the angular frequency (ω) is 37.68 rad/s. Plugging these values into the formula, we can find the mass of the object:

m = 120 N/m / (37.68 rad/s)^2 = 0.088 kg

So, the mass of the object is 0.088 kilograms.