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Law of Conservation of Energy

Q: A small roller coaster starts at point \$A\$ with a speed \$u\$ on a curved track as shown in the figure.The friction between the roller coaster and the track is negligible and its always remains in contact with the track. The speed of roller coaster at point \$D\$ on the track will be:

Let the speed of the roller coaster at point \$D\$ be \$v\$.Applying the law of conservation of energy between points \$A\$ and \$D\$, we get\$\dfrac {1}{2} mu^{2} + mg(2h) = \dfrac {1}{2}mv^{2} + 0\$or \$u^{2} + 4gh = v^{2}\$ or \$v = \sqrt {u^{2} + 4gh}\$.

Q: A ball is made to fall freely from a height.State the kind.kinds of energy possessed by the ball when it is a) at the highest pointb) Just in the middlec) at the ground.

\$a)\$ Potential energy\$b)\$ Potential energy + Kinetic energy\$c)\$ Kinetic energy.

Q: What is a conserved physical quantity?

Physical quantity that remains unchanged in a process is called conserved quantity.

Q: A light spring with spring constant \$1 200 N/m\$ is hung from an elevated support. From its lower end hangs a second light spring, which has spring constant \$1 800 N/m\$. An object of mass \$1.50 kg\$ is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.

The same force makes both light springs stretch.(a) The hanging mass moves down by\$x=x_{1}+x_{2}=\dfrac{mg}{k_{1}}+\dfrac{mg}{k_{2}}=mg\left ( \dfrac{1}{k_{1}}+\dfrac{1}{k_{2}} \right )\$\$=(1.5kg)(9.8m/s^{2})\left ( \dfrac{1}{1200N/m}+\dfrac{1}{1800N/m} \right )\$\$=2.04\times 10^{-2}m\$(b) We define the effective spring constant as\$k=\dfrac{F}{x}=\dfrac{mg}{mg(1/k_{1}+1/k_{2})}=\left ( \dfrac{1}{k_{1}}+\dfrac{1}{k_{2}} \right )^{-1}\$\$=\left ( \dfrac{1}{1200N/m}+\dfrac{1}{1800N/m} \right )=720N/m\$

Q: In the figure given. what is the reading of the balance- \$(g\ = \ 10 N\ kg^{-1})\$

\$F=ma\$\$m=1kg\$\$g=10Nkg^{-1}\$\$F=1 \times 10=10N\$

Q: Write a short note on nuclear energy.

The energy released during the processes of nuclear fission and fusion is called nuclear energy. In both these processes, there is a loss in mass which converts into energy in accordance with Einstein’s mass-energy relation, \$E = mc^2\$. This energy is used in nuclear reactors to produce electrical energy. Energy is produced in the Sun and other stars by nuclear fusion.

Q: A light spring with spring constant \$k_{1}\$ is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant \$k_{2}\$. An object of mass \$m\$ is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system.

(a) The force \$mg\$ is the tension in each of the springs. The bottom of the upper (first) spring moves down by distance \$x_{1} = |F|/k_{1} = mg/k_{1}\$. The top of the second spring moves downby this distance, and the second spring also stretches by \$x_{2} = mg/k_{2}\$. The bottom of the lower spring then moves down by distance\$x_{total}=x_{1}+x_{2}=\dfrac{mg}{k_{1}}+\dfrac{mg}{k_{2}}=mg\left ( \dfrac{1}{k_{1}}+\dfrac{1}{k_{2}} \right )\$(b) From the last equation we have\$mg=\dfrac{x_{1}+x_{2}}{\dfrac{1}{k_{1}}+\dfrac{1}{k_{2}}}\$This is of the form\$|F|=\left ( \dfrac{1}{1/k_{1}+1/k_{2}} \right )(x_{1}+x_{2})\$The downward displacement is opposite in direction to the upward force the springs exert on the load, so we may write \$F = –k_{eff} x_{total}\$, with the effective spring constant for the pair of springs given by\$k_{eff}=\dfrac{1}{1/k_{1}+1/k_{2}}\$