A object-spring system oscillates with an amplitude of 4.0 cm. If the spring constant is 210 N/m and object has a mass of 0.50 kg, determine each of the following values.

(a) the mechanical energy of the system

(b) the maximum speed of the object

(c) the maximum acceleration of the object

To determine the values, we need to use the formulas related to the oscillation of an object-spring system.

(a) The mechanical energy of the system can be calculated using the formula:

E = (1/2)kA^2

where
E = mechanical energy of the system
k = spring constant
A = amplitude of oscillation

Given that the amplitude A is 4.0 cm (which can be converted to meters by dividing by 100), and the spring constant k is 210 N/m, we can substitute these values into the formula to calculate the mechanical energy.

E = (1/2) * 210 N/m * (4.0 cm / 100)^2

Simplifying the equation, we get:

E = (1/2) * 210 N/m * (0.04 m)^2

E = 0.168 J

Therefore, the mechanical energy of the system is 0.168 J.

(b) The maximum speed of the object can be calculated using the formula:

v_max = A * ω

where
v_max = maximum speed of the object
A = amplitude of oscillation
ω = angular frequency

The angular frequency can be calculated using the formula:

ω = √(k/m)

where
k = spring constant
m = mass of the object

Given that the amplitude A is 4.0 cm (which can be converted to meters by dividing by 100), the spring constant k is 210 N/m, and the mass m is 0.50 kg, we can substitute these values into the formula to calculate the angular frequency.

ω = √(210 N/m / 0.50 kg)

Simplifying the equation, we get:

ω = √(420 N/kg)

ω = √(420 m^2/s^2)

Since the units of ω are in rad/s, we can calculate the maximum speed using the formula:

v_max = 4.0 cm / 100 * √(420 m^2/s^2)

v_max = 0.04 m * √(420 m^2/s^2)

v_max ≈ 0.35 m/s

Therefore, the maximum speed of the object is approximately 0.35 m/s.

(c) The maximum acceleration of the object can be calculated using the formula:

a_max = A * ω^2

where
a_max = maximum acceleration of the object
A = amplitude of oscillation
ω = angular frequency

Given that the amplitude A is 4.0 cm (which can be converted to meters by dividing by 100), and the angular frequency ω is √(420 m^2/s^2) as calculated in part (b), we can substitute these values into the formula to calculate the maximum acceleration.

a_max = 4.0 cm / 100 * (√(420 m^2/s^2))^2

a_max = 0.04 m * (420 m^2/s^2)

a_max ≈ 16.8 m/s^2

Therefore, the maximum acceleration of the object is approximately 16.8 m/s^2.

Calculate the angular frequency w. You should know the formula for that. If you don't, review your text.

Let A be the amplitude.

The energy in the sysyem is (1/2) k A^2
The maximum velocity is wA
The maximum acceleration in w^2 A