When a body is executing Simple Harmonic Motion (SHM), its velocity
v at a displacement
x from the mean position can be given by the equation:
v=ω√A2−x2Here,
ω is the angular frequency and
A is the amplitude of the SHM.
Given:
At displacement 4 cm , velocity
10cms−1At displacement 5 cm , velocity
8cms−1We use the equation for velocity at displacement for both given conditions to form two equations:
At
x=4‌cm and
v=10cms−1 :
10=ω√A2−42This can be rewritten as:
10=ω√A2−16At
x=5‌cm and
v=8cms−1 :
8=ω√A2−52This can be rewritten as:
8=ω√A2−25 Now, we square both equations and equate them to form a system of equations:
For the first equation:
100=ω2(A2−16)For the second equation:
64=ω2(A2−25)Divide the first equation by the second equation to eliminate
ω2 :
‌=‌Simplify:
‌=‌Cross-multiplying gives:
25(A2−25)=16(A2−16)Expand and simplify:
25A2−625=16A2−256Rearrange terms:
9A2=369 Therefore:
A2=41The amplitude
A=√41.
We can now use either set of velocity and displacement to find
ω. Using the first set:
‌10=ω√41−16‌10=ω√25Therefore:
‌10=5ω‌ω=2‌rad∕ s The periodic time
T is given by the relation:
T=‌Substitute
ω :
T=‌=π secTherefore, the periodic time t is Option
D:Ï€sec.