The amplitude A=6cm Step 1: Write the energy formulas The total energy is E=‌
1
2
kA2 The potential energy is U=‌
1
2
kx2 The kinetic energy is K=‌
1
2
k(A2−x2) Step 2: Set up the given ratio We want the distance x from the mean position so that ‌
U
K
=‌
4
5
: ‌
‌
1
2
kx2
1
2
k(A2−x2)
=‌
4
5
Step 3: Simplify the equation The ‌
1
2
k cancels out: ‌
x2
A2−x2
=‌
4
5
Step 4: Solve for x2 Multiply both sides by (A2−x2):5x2=4(A2−x2) Expand: 5x2=4A2−4x2 Bring 4x2 to the left: 5x2+4x2=4A2 Combine like terms: 9x2=4A2 Step 5: Find x Divide both sides by 9: x2=‌
4
9
A2 Take the square root: x=‌
2
3
A Since A=6cm : x=‌
2
3
×6=4cm The distance from the mean position is 4 cm .