Question

The figure shows the position-time graph of an object of mass m oscillating on the end of a massless ideal spring of spring constant k. Answer the following questions.

1. Which of the following graphs is the correct velocity-time graph of the oscillation?

2. Which of the following graphs is the correct acceleration-time graph of the oscillation?

3. If the mass of the object is m = 0.500 kg, what is the spring constant k of the ideal spring?

Hint: read o the period of the oscillation T from the graph and use the relations

4. What is the total mechanical energy of the mass-spring system?

Answer 1

1.

T = Time period = π

w = angular frequency = 2π/T = 2π/π = 2

from the position time graph , equation is given as

x = (0.2) Sin(wt +(π/2))

taking derivative both side relative to "t"

dx/dt = (0.2) w Cos(wt +(π/2))

v(t) = (0.4) Cos(wt +(π/2))

2)

v(t) = (0.4) Cos(wt +(π/2))

taking derivative both side relative to "t"

dv(t)/dt = - (0.4)w Sin(wt +(π/2))

a(t) = - (0.8) Sin(wt +(π/2))

3.

w = 2

m = 0.5 kg

k = ?

angular frequency is given as

w = sqrt(k/m)

2 = sqrt(k/0.5)

k = 2 N/m

4.

A = amplitude = 0.2 m

total mechanical energy is given as

E = (0.5) k A²

E = (0.5) (2) (0.2)²

E = 0.04 J