Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ccc}1 2 5 \\ 1 -1 -1 \\ 2 3 -1\end{array}\right]$ Solution:...
Read More →Find the acceleration of the block of mass M in the situation shown in figure.
Question: Find the acceleration of the block of mass $M$ in the situation shown in figure. All the surfaces are frictionless and the pulleys and the string are light. Solution:...
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ll}a b \\ c d\end{array}\right]$, when $(a b-b c) \neq 0$ Solution:...
Read More →Consider the situation shown in figure.
Question: Consider the situation shown in figure. Both the pulleys and the string are light and all the surfaces are frictionless. (a) Find the acceleration of the mass M. (b) Find the tension in the string. (c) Calculate the force exerted by the clamp on the pulley A in the 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. $\left[\begin{array}{cc}2 -3 \\ 4 6\end{array}\right]$ Solution:...
Read More →Find the inverse of each of the matrices given below.
Question: Find the inverse of each of the matrices given below. $\left[\begin{array}{ll}4 1 \\ 2 3\end{array}\right]$ Solution:...
Read More →Calculate the tension in the string shown in figure.
Question: Calculate the tension in the string shown in figure. The pulley and the string are light and all surfaces are frictionless. Take $g$ $=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →In the previous problem,
Question: In the previous problem, suppose $m_{2}=2.0 \mathrm{~kg}$ and $m_{3}=3.0 \mathrm{~kg}$. What should be the mass $\mathrm{m}$ so that it remains at rest? Solution:...
Read More →Solve this
Question: $\left[\begin{array}{cc}3 -5 \\ -1 2\end{array}\right]$ Solution:...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ccc}-1 -2 -2 \\ 2 1 -2 \\ 2 -2 1\end{array}\right]$, show that adj $A=3 A^{\prime}$ Solution:...
Read More →Let m1 = 1 kg, m2 = 2 kg and m, = 3 kg in figure.
Question: Let m1= 1 kg, m2= 2 kg and m, = 3 kg in figure. Find the accelerations of m1, m2 and m3. The string from the upper pulley to m1 is 20 cm when the system is released from rest. How long will it take before m1 strikes the pulley? Solution: $=0.25 \mathrm{sec}$...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ccc}-4 -3 -3 \\ 1 0 1 \\ 4 4 3\end{array}\right]$, show that adj $A=A$ Solution: Thus, adj $A=A$ Hence Proved...
Read More →Find the adjoint of the given matrix and verify in each case that
Question: Find the adjoint of the given matrix and verify in each case that $A .(\operatorname{adj} A)=(\operatorname{adj} A)=m|A| . I$. $\left[\begin{array}{ccc}\cos \alpha -\sin \alpha 0 \\ \sin \alpha \cos \alpha 0 \\ 0 0 1\end{array}\right]$ Solution:...
Read More →In figure m1 = 5 kg, m2 = 2 kg and F = 1N N.
Question: In figure $\mathrm{m} 1=5 \mathrm{~kg}, \mathrm{~m}_{2}=2 \mathrm{~kg}$ and $\mathrm{F}=1 \mathrm{~N} \mathrm{~N}$. Find the acceleration of either block. Describe the motion of $\mathrm{m}$, if the string breaks but F continues to act. Solution:...
Read More →A constant force
Question: A constant force $\mathrm{F}=\mathrm{m}_{2} \mathrm{~g} / 2$ is applied on the block of mass $\mathrm{m}$, as shown in figure. The string and the pulley are light and the surface of the table is smooth. Find the acceleration of $m_{1}$. Solution:...
Read More →Consider the situation shown in figure.
Question: Consider the situation shown in figure. All the surfaces are frictionless and the string and the pulley are light. Find the magnitude of the acceleration of the two blocks. Solution:...
Read More →Figure shows a uniform rod of length
Question: Figure shows a uniform rod of length $30 \mathrm{~cm}$ having a mass of $3.0 \mathrm{~kg}$. The strings shown in the figure are pulled by constant forces of $20 \mathrm{~N}$ and $32 \mathrm{~N}$. Find the force exerted by the $20 \mathrm{~cm}$ part of the rod on the $10 \mathrm{~cm}$ part. All the surfaces are smooth and the strings and the pulleys are light. Solution:...
Read More →Consider the Atwood machine of the previous problem.
Question: Consider the Atwood machine of the previous problem. The larger mass is stopped for a moment $2.0 \mathrm{~s}$ after the system is set into motion. Find the time elapsed before the string is tight again. Solution:...
Read More →In a simple Atwood machine,
Question: In a simple Atwood machine, two unequal masses $\mathrm{m} 1$, and $\mathrm{m} 2$ are connected by a string going over a clamped light smooth pulley. In a typical arrangement $\mathrm{m} 1=300 \mathrm{~g}$ and $\mathrm{m} 2=600 \mathrm{~g}$. The system is released from rest. (a) Find the distance travelled by the first block in the first two seconds. (b) Find the tension in the string. (c) Find the force exerted by the clamp on the pulley. Solution:...
Read More →Prove the following
Question: A force $\vec{F}=\vec{v} \times \vec{A}$ is exerted on a particle in addition to the force of gravity, where $\vec{v}$ is the velocity of the particle and $\vec{A}$ is a constant vector in the horizontal direction. With what minimum speed a particle of mass me projected so that it continues to move undeflected with a constant velocity? Solution:...
Read More →An empty plastic box of mass m is found
Question: An empty plastic box of mass $m$ is found to accelerate up at the rate of $\mathrm{g} / 6$ when placed deep inside water. How much sand should be put inside the box so that it may accelerate down at the rate of $\mathrm{g} / 6 ?$ Solution:...
Read More →The force of buoyancy exerted by the atmosphere
Question: The force of buoyancy exerted by the atmosphere on a balloon is B in the upward direction and remains constant. The force of air resistance on the balloon acts opposite to the direction of velocity and is proportional to it. The balloon carries a mass $M$ and is found to fall down near the earth's surface with a constant velocity v. How much mass should be removed from the balloon so that it may rise with a constant velocity v? Solution:...
Read More →Find the adjoint of the given matrix and verify in each case that
Question: Find the adjoint of the given matrix and verify in each case that $A .(\operatorname{adj} A)=(\operatorname{adj} A)=m|A| . I$. $\left[\begin{array}{lll}4 5 3 \\ 1 0 6 \\ 2 7 9\end{array}\right]$ Solution:...
Read More →Suppose the ceiling in the previous problem
Question: Suppose the ceiling in the previous problem is that of an elevator which is going up with an acceleration of $2.0 \mathrm{~m} / \mathrm{s}^{2}$. Find the elongations. Solution:...
Read More →A block of 2 kg is suspended from the ceiling
Question: A block of $2 \mathrm{~kg}$ is suspended from the ceiling through a mass less spring of spring constant $\mathrm{k}=100 \mathrm{~N} / \mathrm{m}$. What is the elongation of the spring? If another $1 \mathrm{~kg}$ is added to the block, what would be the further elongation? Solution:...
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