OpenResponse Homework Problem 274 An object is attached to a

Open-Response Homework Problem 27.4 An object is attached to a spring in such a way that it undergoes simple harmonic motion with a period of 0.5 s. (a)[3 pts] What is the angular frequency of the oscillations? (b)[3 pts] If the mass of the object is 0.2 kg, what is the value of the spring constant, k? (c)[3 pts] If the amplitude of the oscillations is 10cm, what is the total energy of the spring-object system? (d)[3 pts] Under the assumption above, what is the maximum velocity of the object? (e)[3 pts] What is the maximum acceleration of the object? (f)[3 pts] Write down a formula for the position of the object as a function of time, x(t), assuming that its velocity is maximum at t = 0. Identify the value of all the constants that you use, and sketch the resulting curve.

Solution

Part a :

angular frequeny in SHM is given as :

w = 2*pi / T

where T is time peroid

w = 2 * 3.14 * 0.5

w = 3.14 rad/s

Part B :

angular frequeny in SHM is given as :

w = sqrt(k/m)

where k is spring constant

m is mass of object,

k = w^2 * m

k = 3.14^2 * 0.2

K = 1.97 N/m or 2 n/m (rounded off)

Part c :

x = 10 cm = 0.10 m

Energy, U = 0.5 * k * x^2

U = 0.5 * 1.97 * 0.10^2

U = 0.00985 J or 0.01 J (rounded off)

Part D :

From conervation of energy we have :

POtential energy strored in spring = Kinetic erngy gained by object

0.5 * k * x^2 = 0. 5 * m * v^2

0.5 * 1.97 * 0.10^2 = 0.5 * 0.2 * v^2

v = 0.314 m/s

we are allowed to answer max 4 parts