Question

A spring has natural length $40$ cm and spring constant $500N/m$. A block of mass $1kg$ is attached at one end of the spring and other end of the spring is attached to a ceiling. The block is released from the position, where the spring has length $45cm$:

• A. the block will perform SHM of amplitude $5cm$
• B. the block will have maximum velocity $30\sqrt{5}$ cm/s
• C. the block will have maximum acceleration 15 m/s${}^2$
• D. the minimum elastic potential energy of the spring will be zero

Answer 1

Answer: (B), (C), (D)

The correct options are
B the block will have maximum acceleration 15 m/s${}^2$
C the block will have maximum velocity $30\sqrt{5}$ cm/s
D the minimum elastic potential energy of the spring will be zero

$k{x}_0=mg$
$\therefore$ $x_0=\frac{mg}{k}=\frac{1\times 10}{500}$
$=0.02$ m$=2$ cm
So equilibrium is obtained after an extension of 2 cm of at a length of 42 cm. But it is released from a length of 45 cm.
$\therefore$ A=3 cm=0.03 m
(b) $v_{max}=\omega A=\sqrt{\frac{k}{m}}A$
$=\left(\sqrt{\frac{500}{1}}\right)\left(0.03\right)=0.3\sqrt{5}$ m/s
$=30\sqrt{5}$ cm/s
(c) $a_{max}={\omega }^{2}A$
$=\left(\frac{k}{m}\right)\left(A\right)=\left(\frac{500}{1}\right)\left(0.03\right)=15$ m/s${}^2$
(d) Mean position is at 42 cm length and amplitude is 3 cm. Hence block oscillates between 45 cm length and 39 cm. Natural length 40 cm lies in between these two, where elastic potential energy=0.