A block of mass $m=1\; kg$. moving on a horizontal surface with speed $v_{t}=2 \;m s ^{-1}$ enters a rough patch ranging from $x=0.10\; m$ to $x=2.01\; m .$ The retarding force $F$ on the block in this range is inversely proportional to $x$ over this range,
$F_{r}=\frac{-k}{x}$ for $0.1 < x < 2.01 \;m$
$=0$ for $x < 0.1\; m$ and $x > 2.01\; m$
where $k=0.5\; J .$ What is the final kinetic energy and speed $v_{f}$ of the block as it crosses this patch ?
A block of mass $m=1\; kg$. moving on a horizontal surface with speed $v_{t}=2 \;m s ^{-1}$ enters a rough patch ranging from $x=0.10\; m$ to $x=2.01\; m .$ The retarding force $F$ on the block in this range is inversely proportional to $x$ over this range,
$F_{r}=\frac{-k}{x}$ for $0.1 < x < 2.01 \;m$
$=0$ for $x < 0.1\; m$ and $x > 2.01\; m$
where $k=0.5\; J .$ What is the final kinetic energy and speed $v_{f}$ of the block as it crosses this patch ?
$K_{f}=K_{i}+\int_{0.1}^{2.01} \frac{(-k)}{x} d x$
$=\frac{1}{2} m v_{i}^{2}-\left.k \ln (x)\right|_{0.1} ^{2.01}$
$=\frac{1}{2} m v_{i}^{2}-k \ln (2.01 / 0.1)$
$=2-0.5 \ln (20.1)$
$=2-1.5=0.5 J$
$v_{f}=\sqrt{2 K_{f} / m}=1 m s ^{-1}$
Here, note that In is a symbol for the natural logarithm to the base $e$ and not the logarithm to the base $10 \left[ ln X =\log _{ e } X =2.303 \log _{10} X \right]$
Similar Questions
An adult weighting $600\,N$ raises the centre of gravity of his body by $0.25\,m$ while taking each step of $1\,m$ length in jogging. If he jogs for $6\,km$, calculate the energy utilised by him in jogging assuming that there is no energy loss due to friction of ground and air. Assuming that the body of the adult is capable of converting $10\,\%$ of energy intake in the form of food, calculate the energy equivalents of food that would be required to compensate energy utilised for jogging.
An adult weighting $600\,N$ raises the centre of gravity of his body by $0.25\,m$ while taking each step of $1\,m$ length in jogging. If he jogs for $6\,km$, calculate the energy utilised by him in jogging assuming that there is no energy loss due to friction of ground and air. Assuming that the body of the adult is capable of converting $10\,\%$ of energy intake in the form of food, calculate the energy equivalents of food that would be required to compensate energy utilised for jogging.
$A$ particle with constant total energy $E$ moves in one dimension in a region where the potential energy is $U(x)$. The speed of the particle is zero where
$A$ particle with constant total energy $E$ moves in one dimension in a region where the potential energy is $U(x)$. The speed of the particle is zero where
A ball is dropped from a height of $20\, cm$. Ball rebounds to a height of $10\, cm$. What is the loss of energy ? ................ $\%$
A ball is dropped from a height of $20\, cm$. Ball rebounds to a height of $10\, cm$. What is the loss of energy ? ................ $\%$
Two trolleys of mass m and $3m$ are connected by a spring. They were compressed and released once, they move off in opposite direction and comes to rest after covering distances ${S_1}$and ${S_2}$ respectively. Assuming the coefficient of friction to be uniform, the ratio of distances ${S_1}:{S_2}$ is
Two trolleys of mass m and $3m$ are connected by a spring. They were compressed and released once, they move off in opposite direction and comes to rest after covering distances ${S_1}$and ${S_2}$ respectively. Assuming the coefficient of friction to be uniform, the ratio of distances ${S_1}:{S_2}$ is
If the increase in the kinetic energy of a body is $22\%$, then the increase in the momentum will be ........... $\%$
If the increase in the kinetic energy of a body is $22\%$, then the increase in the momentum will be ........... $\%$