Question

A horizontal object-spring system oscillates with an amplitude of 2.8 cm. If the spring constant is 275 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
m/s
(c) the maximum acceleration of the object
m/s2

Answer 1

Answer:

(a) the mechanical energy of the system, U = 0.1078 J

(b) the maximum speed of the object, Vmax = 0.657 m/s

(c) the maximum acceleration of the object, a_max = 15.4 m/s²

Explanation:

Given;

Amplitude of the spring, A = 2.8 cm = 0.028 m

Spring constant, K = 275 N/m

Mass of object, m = 0.5 kg

(a) the mechanical energy of the system

This is the potential energy of the system, U = ¹/₂KA²

U = ¹/₂ (275)(0.028)²

U = 0.1078 J

(b) the maximum speed of the object

$V_{max} =\omega*A= \sqrt{\frac{K}{M} } *A\\\\V_{max} = \sqrt{\frac{275}{0.5} } *0.028\\\\V_{max} = 0.657 \ m/s$

(c) the maximum acceleration of the object

$a_{max} = \frac{KA}{M} \\\\a_{max} = \frac{275*0.028}{0.5}\\\\a_{max} = 15.4 \ m/s^2$

Answer 2

The characteristics of the simple harmonic motion allows to find the results for the questions of the oscillating mass are:

     a) The total energy is: Em = 0.1078 J

     b) The maximum speed is: v = 0.657 m / s

    c) the maximum acceleration is: a = 15.4 m / s²

Given parameters

• The amplitude A = 2.8 cm = 2.8 10⁻² m

• The spring constant k = 275 N / m

• Mass m = 0.50 kg

To find

    a) Mechanical energy

    b) Maximum speed

    c) Maximum acceleration

the simple harmonic movement is an oscillatory movement where the restoring force is proportional to the displacement, it is described by the expression:

         x = A cos (wt + Ф)

         w² = k / m

Where x is the displacement, A the amplitude w the angular velocity, t the time, Ф a phase constant, k the spring constant and m the mass.

A) The mechanical energy is

        Em = ½ k A²

Let's calculate.

       Em = ½ 275 (2.8 10⁻²) ²

       Em = 0.1078 J

b) Velocity is defined as the change of position with respect to time.

       v = $\frac{dx}{dt}$  = - Aw sin ( wt + fi)

To obtain the maximum velocity, the sine function must be ±1

        $v_{max}$ = w A

         

Let's calculate

        w = $\sqrt{\frac{275}{0.5} }$  

        w = 23.45 rad / s

        $v_{max}$ = 23.45 2.8 10⁻²

        $v_{max}$ = 0.657 m / s

c) maximum acceleration.

Acceleration is defined as the change in velocity withtrspect to time.

       a = $\frac{dv}{dt}$  = - A w² cos (wt + fi)

To have the maximum value, the cosine function must be maximum, that is ±1

      a = A w²

 

let's calculate

       a = 2.8 10⁻²  23.45²

       a = 15.4 m / s²

In conclusion using the characteristics of the simple harmonic motion we can find the results for the questions of the oscillating mass are:

     a) The total energy is:  Em = 0.1078 J

     b) The maximum speed is:  v = 0.657 m / s

     c) the maximum acceleration is: a = 15.4 m / s²

Learn more here:  brainly.com/question/17315536