Question

║⊕QUESTION⊕║
“Energy is liberated matter, the matter is energy waiting to happen.”
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CLASS 11
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Show that the total energy is a constant for an oscillating loaded spring and draw its
1. force-displacement graph
2. energy - displacement graph
3. energy-time graph

Answer 1

◆【Solution】◆

For a particle of mass m in SHM. there is one simple condition that Force is directly proportional to negative of displacement.

F = -kx

(1) Force displacement graph

Let F is y axis

and displacement is X axis than equation look like

y = -kx

In this equation -k is slope of the line with zero intercept.

For SHM

X = Asin(wt)

v = Awcos(wt)

where A is maximum displacement

w is angular frequency = 2π/t

than we know that potential energy is du

du = -f.dx

du = -(-kx.dx)

du = (k)x.dx

U = kx²/2

U = k(Asinwt)²/2

U = kA²Sin²(wt)/2

we know that k = mw²

U = mw²A²Sin²wt/2.....(1)

We know that Kinetic energy is 1/2mv²

K = (1/2)m(A.w.coswt)²

K = mA²w²cos²wt/2......(2)

Add (1) and (2)

T.E = mw²A²Sin²wt/2 + 1/2m(A.w.coswt)²

T.E = mw²A²/2 (Sin²wt + Cos²wt)

Total energy = mw²A²/2

Hence total energy is independent of displacement and Time so graph of both will remain constant and be a parallel line with Total energy axis.

In this energy-time graph line parabola is for kinetic energy and blue one is for potential energy and black one is for total energy.

the same graph can be used for energy - displacement

Answer 2

Answer:

Let f:Z↦Z be defined as f(x)=x

2

,x∈Z.

We know that the square of an integer is always a unique integer.

So, ''f'' is a function.

Now, since f(−2)=f(2)=4, ''f'' is not an injection.

There is no integer x∈Z:f(x)=−1.

Hence , ''f'' is not a surjection.

Since ''f'' is neither one-one nor onto, ''f'' is not a bijection.