Five different experiments are carried out. In each experiment, a block is attac

Question

Five different experiments are carried out. In each experiment, a block is attached to a horizontal spring. The block is pulled back a certain distance and released. The block oscillates back and forth on a frictionless surface. Rank the amplitude of oscillation for each of the following situations. (Rank the smallest amplitude as 1).
1 2 3 4 5  A block of mass M is attached to a spring with a spring constant k, pulled back a distance (1/2)d, and released.
1 2 3 4 5  A block of mass M is attached to a spring with a spring constant (1/2)k, pulled back a distance 2d, and released.
1 2 3 4 5  A block of mass M is attached to a spring with a spring constant 2k, pulled back a distance d, and released.
1 2 3 4 5  A block of mass M is attached to a spring with a spring constant k, pulled back a distance d, and released.
1 2 3 4 5  A block of mass 2M is attached to a spring with a spring constant k, pulled back a distance d, and released.

Explanation / Answer

1.

Potential energy of the spring, U = 0.5kA2

0.5Mv2 = 0.5kA2

Spring constant is inversely proportional to the square of the amplitude.

Mass is directly propotional to square of the amplitude.

A2 = sqrt{[m2k1 / m1k2]}A1

If k2 = k1 = k, then A2 =  sqrt{[k / k]}A1 = A1

2.

If k2 = 1/2 k = 1/2 k1 and M1=M2=M, then A2 =  sqrt{[2Mk / Mk]}A1 = [1.414]A1

3.

If k2 = 2 k = 2 k1, and M1=M2=M, then A2 =  sqrt{[Mk / 2Mk]}A1 = [0.707]A1

4. c

If k2 = k = k1,and M1=M2=M, then A2 =  sqrt{[Mk / Mk]}A1 = A1

5.

If k2 = k = k1,and M2=2M1 = 2M, then A2 =  sqrt{[2Mk / Mk]}A1 = [1.414]A1

Ranking: Small to large: 3 < 1=4 < 2=5

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