Find the acceleration of a particle placed on the surface
Question: Find the acceleration of a particle placed on the surface of the earth at the equator due to earth's rotation. The diameter of earth $=12800 \mathrm{~km}$ and it takes 24 hours for the earth to complete one revolution about its axis. Solution:...
Read More →Find the acceleration of the moon with respect to
Question: Find the acceleration of the moon with respect to the earth from the following data: Distance between the earth and the moon $=3.85 \times 10^{5} \mathrm{~km}$ and the time taken by the moon to complete one revolution around the earth $=27.3$ days. Solution:...
Read More →Figure shows a small block of mass
Question: Figure shows a small block of mass $m$ kept at the left end of a larger block of mass Mand length I. The system can slide on a horizontal road. The system is started towards right with an initial velocity v. The friction coefficient between the road and the bigger block is $\mu$ and that between the block is $\mu / 2$. Find the time elapsed before the smaller blocks separates from the bigger block. Solution: FBD of mass m,...
Read More →A person (40 kg) is managing to be at rest between two vertical
Question: A person $(40 \mathrm{~kg})$ is managing to be at rest between two vertical walls by pressing one wall A by his hands and feet and the other wall Bby his back. Assume that the friction coefficient between his body and the walls is $0.8$ and that limiting friction acts at all the contacts. (a) Show that the person pushes the two walls with equal force. (b) Find the normal force exerted by either wall on the person. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →Prove that
Question: Prove that: (i) adj $\mathrm{I}=\mathrm{I}$ (ii) adj $\mathrm{O}=\mathrm{O}$ (iii) $\mathrm{I}^{-1}=\mathrm{I}$. Solution:...
Read More →Solve this following
Question: Show that the $A=\left[\begin{array}{ccc}1 0 -2 \\ -2 -1 2 \\ 3 4 1\end{array}\right]$ satisfies the equation $A^{3}-A^{2}-3 A-I=O$, and hence find $A^{-1}$. Solution:...
Read More →A block of mass 2 kg is pushed against a rough vertical wall with
Question: A block of mass $2 \mathrm{~kg}$ is pushed against a rough vertical wall with a force of $40 \mathrm{~N}$, coefficient of static friction being $0.5$. Another horizontal force of $15 \mathrm{~N}$, is applied on the block in a direction parallel to the wall. Will the block move? If yes, in which direction? If no, find the frictional force exerted by the wall on the block. Solution:...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ll}3 -2 \\ 4 -2\end{array}\right]$. Find the value of $\lambda$ so that $A^{2}=\lambda A-21 .$ Hence, find $A^{-1}$. Solution:...
Read More →Find the acceleration of the block of mass
Question: Find the acceleration of the block of mass $M$ in the situation of figure. The coeffiicient of friction between the two blocks is $\mu 1$, and that between the bigger block and the ground is $\mu 2$. Solution: When a block $M$, moves with acceleration a towards right block $m$ moves downwards and rightwards with acceleration $2 a$ and a respectively. Drawing FBD of M mass...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ll}3 1 \\ 7 5\end{array}\right]$, find $\square$ and $\square$ such that $A^{2}+\square I=0$ A. Hence, find $A^{-1}$ Solution:...
Read More →Solve this following
Question: If $A=\left[\begin{array}{cc}-1 -1 \\ 2 -2\end{array}\right]$, show that $A^{2}+3 A+4 I_{2}=0$ and hence find $A^{-1}$ Solution: Ans. $\mathrm{A}^{-1}=\left[\begin{array}{cc}-\frac{1}{2} \frac{1}{4} \\ -\frac{1}{2} -\frac{1}{4}\end{array}\right]$...
Read More →A block of mass $m$ slips on a rough horizontal
Question: A block of mass $m$ slips on a rough horizontal table under the action of a horizontal force applied to it. The coefficient of friction between the block and the table is $\mathrm{g}$. The table does not move on the floor. Find the total frictional force applied by the floor on the legs of the table. Do you need the friction coefficient between the table and the floor or the mass of the table? Solution:...
Read More →Consider the situation shown in figure.
Question: Consider the situation shown in figure. Suppose a small electric field Eexists in the space in the vertically upward direction and the upper block carries a positive charge $\mathrm{Q}$ on its top surface. The friction coefficient between the two blocks is $g$ but the floor is smooth. What maximum horizontal force Fcan be applied without disturbing the equilibrium? [Hint: The force on a charge $Q$ by the electric field $E$ is $F=Q E$ in the direction of $E$.] Solution:...
Read More →Suppose the entire system of the previous question is
Question: Suppose the entire system of the previous question is kept inside an elevator which is coming down with an acceleration a g. Repeat parts (a) and (b). Solution: In elevator, moving down with acceleration a, effective acceleration will be $g_{e f f}=g-a$ So, replace $g$ by $g_{\text {eff }}$ in above answers....
Read More →Two masses M_1 and M_2 are connected by a light rod
Question: Two masses $M_{1}$ and $M_{2}$ are connected by a light rod and the system is slipping down a rough incline of angle $\theta$ with the horizontal. The friction coefficient at both the contacts is $\mu$. Find the acceleration of the system and the force by the rod on one of the blocks. Solution:...
Read More →Figure shows two blocks in contact sliding down
Question: Figure shows two blocks in contact sliding down an inclined surface of inclination $30^{\circ}$. The friction coefficient between the block of mass $2.0 \mathrm{~kg}$ and theincline is $\mathrm{p}$ and that between the block of mass $4.0 \mathrm{~kg}$ and the incline is $\mu$. Calculate the acceleration of the $2.0 \mathrm{~kg}$ block if $(\mathrm{a}) \mu 1=0.20$ and $\mu 2=0.30$, (b) $\mu 1=0.30$ and $\mu 2=0.20$. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →Show that the matrix
Question: Show that the matrix $A=\left[\begin{array}{cc}-8 5 \\ 2 4\end{array}\right]$ satisfies the equation $\square^{2}+4 \square-42=0$ and hence find $A^{-1}$. Solution:...
Read More →A car starts from rest on a half kilometre long bridge.
Question: A car starts from rest on a half kilometre long bridge. The co-efficient of friction between the tyre and the road is $1.0$. Show that one cannot drive through the bridge is less than 10 s. Solution:...
Read More →A car is going at a speed of
Question: A car is going at a speed of $21.6 \mathrm{~km} / \mathrm{hr}$ when it encounters a $12.8 \mathrm{~m}$ long slope of angle $30^{\circ}$. The friction co-efficient between the road and the tyre is $\frac{\sqrt{3}}{2}$. Show that no matter how hard the driver applies the brakes, the car will reach the bottom with a speed greater than $36 \mathrm{~km} / \mathrm{hr}$. Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →The friction coefficient between an athelete's shoes
Question: The friction coefficient between an athelete's shoes and the ground is $0.90$. Suppose a superman wears these shoes and races for $50 \mathrm{~m}$. There is no upper limit on his capacity of running at high speeds. (a) Find the minimum time that he will have to take in completingthe $50 \mathrm{~m}$ starting from rest. (b) Suppose he takes exactly this minimum time to complete the $50 \mathrm{~m}$, what minimum time will he take to stop? Solution:...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ll}3 2 \\ 2 1\end{array}\right]$, verify that $A^{2}-4 A-I=O$, and hence find $A^{-1}$ Solution: $=0$ $=\mathrm{RHS}$...
Read More →A body starts slipping down an incline and
Question: A body starts slipping down an incline and moves half meter in half second. How long will it take to move the next half meter? Solution:...
Read More →Obtain the inverses of the matrices
Question: Obtain the inverses of the matrices $\left[\begin{array}{ccc}1 \mathrm{p} 0 \\ 0 1 \mathrm{p} \\ 0 0 1\end{array}\right]$ and $\left[\begin{array}{lll}1 0 0 \\ \mathrm{q} 1 0 \\ 0 \mathrm{q} 1\end{array}\right]$. And, hence find the inverse of the matrix $\left[\begin{array}{ccc}1+\mathrm{pq} \mathrm{p} 0 \\ \mathrm{q} 1+\mathrm{pq} \mathrm{p} \\ 0 \mathrm{q} 1\end{array}\right]$ Solution:...
Read More →In a children-park an inclined plane is constructed
Question: In a children-park an inclined plane is constructed with an angle of incline $45^{\circ}$ in the middle part. Find the acceleration of a boy sliding on it if the friction coefficient between the cloth of the boy and the incline is $0.6$ and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution:...
Read More →A body of mass 2 kg is lying on a rough inclined plane of inclination
Question: A body of mass $2 \mathrm{~kg}$ is lying on a rough inclined plane of inclination $30^{\circ}$. Find the magnitude of the force parallel to the incline needed to make the block move (a) up the incline (b) down the incline. Coefficient of static friction = $0.2$. Solution: (a) (b)...
Read More →