A mass m is attached to a spring of constant k and is undergoing SHM. The total energy of the system is proportional to:

March 20, 2026Physics

ScanSolve AI Answer

March 20, 2026

Answer

(2) and (4)

Step 1: Displacement in SHM
$x = A \cos(\omega t)$
where $\omega = \sqrt{\frac{k}{m}}$ .

Step 2: Velocity
$v = \frac{dx}{dt} = -A \omega \sin(\omega t)$ .

Step 3: Kinetic energy
$K = \frac{1}{2} m v^{2} = \frac{1}{2} m (A \omega \sin(\omega t))^{2} = \frac{1}{2} m \omega^{2} A^{2} \sin^{2}(\omega t).$

Step 4: Substitute $\omega^{2} = \frac{k}{m}$
$K = \frac{1}{2} m \left(\frac{k}{m}\right) A^{2} \sin^{2}(\omega t) = \frac{1}{2} k A^{2} \sin^{2}(\omega t).$

Step 5: Potential energy
$U = \frac{1}{2} k x^{2} = \frac{1}{2} k (A \cos(\omega t))^{2} = \frac{1}{2} k A^{2} \cos^{2}(\omega t).$

Step 6: Total energy
$E = K + U = \frac{1}{2} k A^{2} \sin^{2}(\omega t) + \frac{1}{2} k A^{2} \cos^{2}(\omega t) = \frac{1}{2} k A^{2} (\sin^{2}(\omega t) + \cos^{2}(\omega t)).$

Step 7: Simplify using trig identity $\sin^{2} \theta + \cos^{2} \theta = 1$
$E = \frac{1}{2} k A^{2}.$

Step 8: Proportionality
$E \propto k$ and $E \propto A^{2}$ .
$E$ does not depend on $m$ , and is not proportional to $A$ .

Final answer: $\boxed{(2) and (4)}$

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Quick Answer

Displacement in SHM x = A ( t) where = sqrt((k)/(m)). Step 2: Velocity v = (dx)/(dt) = -A ( t).

Asked on March 20, 2026Physics

This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

ScanSolve AI AnswerAnswered on March 20, 2026

Step 1: Displacement in SHM x = A ( t) where = sqrt((k)/(m)). Step 2: Velocity v = (dx)/(dt) = -A ( t). Step 3: Kinetic energy K = (1)/(2) m v^2 = (1)/(2) m (A ( t))^2 = (1)/(2) m ^2 A^2 ^2( t). Step 4: Substitute ^2 = (k)/(m) K = (1)/(2) m ((k)/(m)) A^2 ^2( t) = (1)/(2) k A^2 ^2( t). Step 5: Potential energy U = (1)/(2) k x^2 = (1)/(2) k (A ( t))^2 = (1)/(2) k A^2 ^2( t). Step 6: Total energy E = K + U = (1)/(2) k A^2 ^2( t) + (1)/(2) k A^2 ^2( t) = (1)/(2) k A^2 (^2( t) + ^2( t)). Step 7: Simplify using trig identity ^2 + ^2 = 1 E = (1)/(2) k A^2. Step 8: Proportionality E k and E A^2. E does not depend on m, and is not proportional to A. Final answer: (2) and (4)