A block of mass m is kept on a horizontal table.
Question: A block of mass $m$ is kept on a horizontal table. If the static friction coefficient is $\mu$, find the frictional force acting on the block. Solution: Since no driving force is present to move the block. So, frictional force will be zero....
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Question: Compute $(\mathrm{AB})^{-1}$ when $\mathrm{A}=\left[\begin{array}{ccc}1 1 2 \\ 0 2 -3 \\ 3 -2 4\end{array}\right]$ and $\mathrm{B}^{-1}=\left[\begin{array}{ccc}1 2 0 \\ 0 3 -1 \\ 1 0 2\end{array}\right]$ Solution:...
Read More →A block is projected along a rough horizontal road
Question: A block is projected along a rough horizontal road with a speed of $10 \mathrm{~m} / \mathrm{s}$. If the coefficient of kinetic friction is $0.10$, how far it will travel before coming to rest? Solution:...
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Question: If $A=\left[\begin{array}{cc}9 -1 \\ 6 -2\end{array}\right]$ and $B=\left[\begin{array}{cc}-4 3 \\ 5 -4\end{array}\right]$, verify that $(A B)^{-1}=B^{-1} A^{-1}$ Solution:...
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Question: If $A=\left[\begin{array}{ll}3 2 \\ 7 5\end{array}\right]$ and $B=\left[\begin{array}{ll}6 7 \\ 8 9\end{array}\right]$, verify that $(A B)^{-1}=B^{-1} A^{-1}$ Solution:...
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Question: Let $D=\operatorname{diag}\left[d_{1}, d_{2}, d_{3}\right]$, where none of $d_{1}, d_{2}, d_{3}$ is 0 ; prove that $D^{-1}=\operatorname{diag}\left[d_{1}^{-1}, d_{2}^{-1}, d_{3}^{-1}\right]$. Solution:...
Read More →A block is kept on the floor of an elevator at rest.
Question: A block is kept on the floor of an elevator at rest. The elevator starts descending with an acceleration of $12 \mathrm{~m} / \mathrm{s}^{2}$. Find the displacement of the block during the first $0.2 \mathrm{~s}$ after the start. Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
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Question: If $A=\frac{1}{9}\left[\begin{array}{ccc}-8 1 4 \\ 4 4 7 \\ 1 -8 4\end{array}\right]$ show that $A^{-1}=A^{\prime}$. Solution:...
Read More →A car is speeding up on a horizontal road
Question: A car is speeding up on a horizontal road with an acceleration a. Consider the following situations in the car. (i) A ball is suspended from the ceiling through a string and is maintaining a constant angle with the vertical. Find this angle. (ii) A block is kept on a smooth incline and does not slip on the incline. Find the angle of the incline with the horizontal. Solution: (1) (2)...
Read More →A block A can slide on a frictionless incline of
Question: A block A can slide on a frictionless incline of angle 0 and length 1 , kept inside an elevator going up with uniform velocity v. Find the time taken by the block to slide down the length of the incline if it is released from the top of the incline. Solution:...
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Question: If $A=\left[\begin{array}{rrr}3 -3 4 \\ 2 -3 4 \\ 0 -1 1\end{array}\right]$, prove that $A^{-1}=A^{3}$ Solution:...
Read More →Figure shows a man of mass
Question: Figure shows a man of mass $60 \mathrm{~kg}$ standing on a light weighing machine kept in a box of mass $30 \mathrm{~kg}$. The box is hanging from a pulley fixed to the ceiling through a light rope, the other end of which is held by the man himself. If the man manages to keep the box at rest, what is the weight shown by the machine? What force should he exert on the rope to get his correct weight on the machine? Solution: (A) (B) $a=3 g$ $\mathrm{T} 1=60 \mathrm{a}=60 \times 3 \mathrm{...
Read More →The monkey B shown in figure holding on to the tail
Question: The monkey B shown in figure holding on to the tail of the monkey A which is climbing up a rope. The masses of the monkeys $\mathrm{A}$ and $\mathrm{B}$ are $5 \mathrm{~kg}$ and $2 \mathrm{~kg}$ respectively. If A can tolerate a tension of $30 \mathrm{~N}$ in its tail, what force should it apply on the rope in order to carry the monkey B with it? Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: $T-5 g-30=5 a$ $30-2 g=2 a$ $\mathrm{T}=25+50+30=105 \mathrm{~N}=\mathrm{T}_{\ma...
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Question: If $A=\left[\begin{array}{ccc}1 -1 1 \\ 2 -1 0 \\ 1 0 0\end{array}\right]$, show that $A^{-1}=A^{2}$ Solution: Hence Proved...
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Question: If $A=\left[\begin{array}{cc}2 3 \\ 5 -2\end{array}\right]$, show that $A^{-1}=\frac{1}{19} A$. Solution: Hence Proved...
Read More →A monkey is climbing on a rope that goes over
Question: A monkey is climbing on a rope that goes over a smooth light pulley and supports a block of equal mass at the other end. Show that whatever forces the monkey exerts on the rope, the monkey and the block move in the same direction with equal acceleration. If initially both were at rest, their separation will not change as time passes. Solution: That means both are in same direction....
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ccc}8 -4 1 \\ 10 0 6 \\ 8 1 6\end{array}\right]$ Solution:...
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ccc}2 -1 4 \\ -3 0 1 \\ -1 1 2\end{array}\right]$ Solution: Ans. $\frac{1}{19} \cdot\left[\begin{array}{ccc}1 -6 1 \\ -5 -8 14 \\ 3 1 3\end{array}\right]$...
Read More →A monkey of mass
Question: A monkey of mass $15 \mathrm{~kg}$ is climbing on a rope with one end fixed to the ceiling. If it wishes to go up with an acceleration of 1 $\mathrm{m} / \mathrm{s}^{2}$, how much force should it apply to the rope? If the rope is $5 \mathrm{~m}$ long and the monkey starts from rest, how much time will it take to reach the ceiling? Solution:...
Read More →Find the acceleration of the 500 g block in figure
Question: Find the acceleration of the $500 \mathrm{~g}$ block in figure Solution:...
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ccc}0 0 -1 \\ 3 4 5 \\ -2 -4 -7\end{array}\right]$ Solution:...
Read More →Find the acceleration of the blocks A
Question: Find the acceleration of the blocks $A$ and $B$ in the three situations shown in figure. Solution:...
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ccc}2 -3 3 \\ 2 2 3 \\ 3 -2 2\end{array}\right]$ Solution: Ans. $\frac{1}{5} \cdot\left[\begin{array}{ccc}-2 0 3 \\ -1 1 0 \\ 2 1 -2\end{array}\right]$...
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ccc}2 -1 1 \\ 3 0 -1 \\ 2 6 0\end{array}\right]$ Solution: We have, $A=\left[\begin{array}{ccc}2 -1 1 \\ 3 0 -1 \\ 2 6 0\end{array}\right]$ Ans. $\frac{1}{32} \cdot\left[\begin{array}{ccc}6 6 1 \\ -2 -2 5 \\ 8 -14 3\end{array}\right]$...
Read More →Find the mass M of the hanging block in figure
Question: Find the mass M of the hanging block in figure which will prevent the smaller block from slipping over the triangular block. All the surfaces are frictionless and the strings and the pulleys are light. Solution:...
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