In each of the cases described below, a block oscillates in SHM at the end of a
ID: 2270066 • Letter: I
Question
In each of the cases described below, a block oscillates in SHM at the end of a spring on a frictinless, horizontal surface.
Case1: The block has mass m, the spring constant is k, and the amplitude is A.
Case2: The block has mass m, the spring constant is k, and the amplitude is 2A.
Case3: The block has mass 2m, the spring constant is k, and the amplitude is A.
Case4: The block has mass m, the spring constant is 2k, and the amplitude is A.
(a) Rank the four cases from largest to smallest in terms of their periods of oscillation, T. Indicate if any of the periods are equal. Briefly justify your ranking.
(b) Rank the four cases from largest to smallest in terms of their maximum speed, Vmax. Indicate if any of the maximum speeds are equal. Briefly justify your ranking.
(c) Rank the four cases from largest to smallest in terms of their total energy, E. Indicate if any of the total energies are equal. Briefly justify your ranking.
Explanation / Answer
a) The period T doesn't depend on the amplitude A:
T = 2xPIxSQRT(m/k)
Where PI is the number: 3.141492...
T = 6.28 SQRT(m/k)
The bigger is m the bigger is the period and the bigger the k the smaller is the period.
The greater value of T is for the minimum k and maximum 2m, case 3
The second smaller value of T is for cases 1 and 2 (same masses and constant k)
The smallest value of T is for the maximum 2k and minimum m, case 4.
The rank is:
Case 3, cases 1 and 2, case 4.
b) The maximum kinetic energy corresponds to the maximum total energy E of the SHM.
E = (1/2).k.(A)^2
(1/2).m.V^2 = (1/2).k.(A)^2
V = A.SQRT(k/m)
The maximum values of A and SQRT(k/m) correspond to the maximum speed:
The rank is given by:
Case 1: V1 = A. SQRT(k/m)
Case 2: V2 = 2A. SQRT(k/m)
Case 3: V3= A. SQRT[k/(2m)]
Case 4: V4 = A. SQRT(2k/m)
V2/v1 = 2 =====>V2 = 2V1
V2/V3 = 2. SQRT(2)======> V2 = 2.SQRT(2)V3
V2/V4 = 2/SQRT(2)=======> V2 = [2/SQRT(2)]V4
V2>V4>V1>V3
c) E = 1/2 k A^2
1: 1/2 kA^2
2: 2 k A^2
3: 1/2 k a^2
4: k A^2
2>1=3>4
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