Magnitude using Vector: Derivation

Q.

A mass M is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when?

1. the mass is at the highest point

2. the wire is horizontal

3. the mass is at the lowest point

4. inclined at an angle of 60° from vertical

Q. The motion of a mass on a spring, with spring constant $K$ is as shown in figure.


The equation of motion is given by $x\left(t\right)=A\sin \omega t+B\cos \omega t$ with $\omega =\sqrt{\frac{K}{m}}$.

Suppose that at time $t=0,$ the position of mass is $x\left(0\right)$ and velocity $v\left(0\right),$ then its displacement can also be represented as $x\left(t\right)=C\cos (\omega t-\varphi ),$ where $C$ and $\varphi$ are :

1. $C=\sqrt{\frac{v(0{)}^2}{{\omega }^2}+x(0{)}^2,}\varphi ={\tan }^{-1}\left(\frac{v\left(0\right)}{x\left(0\right)\omega }\right)$
2. $C=\sqrt{\frac{2v(0{)}^2}{{\omega }^2}+x(0{)}^2},\varphi ={\tan }^{-1}\left(\frac{x\left(0\right)\omega }{2v\left(0\right)}\right)$
3. $C=\sqrt{\frac{2v(0{)}^2}{{\omega }^2}+x(0{)}^2,}\varphi ={\tan }^{-1}\left(\frac{x\left(0\right)\omega }{v\left(0\right)}\right)$
4. $C=\sqrt{\frac{2v(0{)}^2}{{\omega }^2}+x(0{)}^2,}\varphi ={\tan }^{-1}\left(\frac{v\left(0\right)}{x\left(0\right)\omega }\right)$

Q. A chain of $125$ links is $1.25m$ long and has a mass of $2\text{kg}$. With the ends fastened together, it is set rotating at $3000\text{rpm}$. The centripetal force(in $\text{N}$) on each link is( consider $\pi =3.14$)

Q.

Repeat the previous exercise if the angle between each pair of springs is ${120}^{\circ }$ initially.

Q. A particle of mass $m$ is attached to three identical springs $\text{A, B}$ and $\text{C}$ each of force constant $k$ as shown in figure. If the particle of mass $m$ is pushed slightly against the spring $\text{A}$ and released, then the time period of oscillation is

1. $2\pi \sqrt{\frac{2m}{k}}$
2. $2\pi \sqrt{\frac{m}{2k}}$
3. $2\pi \sqrt{\frac{m}{k}}$
4. $2\pi \sqrt{\frac{m}{3k}}$

Q. A rod AB of mass $M$ and length $L$ is kept on a smooth horizontal surface. A particle of mass $m$ moving with speed $v_0$ collides with the stationary rod (shown in figure) at point C, at $\frac{L}{4}$ distance from the centre of the rod. After collision, the particle becomes stationary, but the rod translates with speed $v$ and rotates anticlockwise with angular speed $\omega$ about the centre of the rod O. Following conservation of angular momentum, which of the following statements is correct for the (rod + particle) system?

1. About point O, $m{v}_0\frac{L}{4}=\frac{M{L}^2}{12}.\omega$
2. About point C, $Mv.\frac{L}{4}=\frac{M{L}^2}{12}.\omega$
3. About point A, $m{v}_0.\frac{3L}{4}=Mv.\frac{L}{2}+\frac{M{L}^2}{12}.\omega$
4. About point B, $m{v}_0.\frac{3L}{4}=Mv.L+\frac{M{L}^2}{12}.\omega$

Q.

A person stands on a spring balance at the equator. (a) By what fraction is the balance reading less than his true weight ? (b) If the speed of earth's rotation is increased by such an amount that the balance reading is half the true weight, what will be the length of the day in this case ?

Q. The angular velocity of a body changes from one revolution per $9$ second to $1$ revolution per second. The ratio of its radius of gyration in the two cases is
(Assume that net torque applied is zero)

1. $1:9$
2. $3:1$
3. $9:1$
4. $1:3$

Q.

A particle of mass m is attached to M via a massless string passing through a hole O in the horizontal table as shown, Mass M is kept stationary whereas mass m is rotating in a circle of radius r with angular speed $\omega$.

1. $Mg=mr{\omega }^2$

2. $Mg>mr{\omega }^2$

3. $Mg\ge mr{\omega }^2$

4. $Mg<mr{\omega }^2$

Q. A string can withstand a tension of $25\text{N}$. What is the greatest speed at which a body of mass $1\text{kg}$ can be whirled in a horizontal circle using a $1\text{m}$ length of a string ?

1. $2.5{\text{ms}}^{-1}$
2. $5.0{\text{ms}}^{-1}$
3. $7.5{\text{ms}}^{-1}$
4. $10{\text{ms}}^{-1}$

Q. A rod of length $L$ is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. The value of $x$ for point $P$, where tension is ${\left(\frac{1}{4}\right)}^{th}$ of the tension at $O$

1. $\frac{L}{2}$
2. $\frac{L}{4}$
3. $\frac{\sqrt{3}L}{2}$
4. $\frac{3L}{4}$

Q. A coin placed on a gramophone record rotating at $33\text{rpm}$ flies off record, if it is placed at a distance of more than $16\text{cm}$ from the axis of rotation. If the record is revolving at $66\text{rpm}$, the coin will fly off if is placed at a distance( in $\text{cm}$) not less than

Q. A point of mass is tied to one end of a cord whose other end is passes through a vertical hollow tube, caught in one hand. The point mass is being rotated in a horizontal circle of radius $2\text{m}$ with speed of $4\text{m/s}$. The cord is then pulled down so that the radius of the circle reduces to $1\text{m}$. Find the ratio of final kinetic energy and initial kinetic energy.

1. $1:1$
2. $3:1$
3. $2:1$
4. $4:1$

Q. An object of $60\text{kg-wt}$ is weighed by hanging it to the spring balance, from the ceiling of a car. If the car starts moving around a curve of radius $10\text{m}$ at a speed $10{\text{ms}}^{-1}$. The reading of the spring balance would be

1. $60\sqrt{2}\text{kg-wt}$
2. $60\text{kg-wt}$
3. $120\sqrt{2}\text{kg-wt}$
4. $600\text{kg-wt}$

Q. A car of mass $m=1000\text{kg}$ is moving with constant speed $v=10\text{m/s}$ on a parabolic shaped bridge $AFOE$ of span $l=40\text{m}$ and height $h=20\text{m}$ as shown in the figure. Then the net force applied by the bridge on the car, when it is at point $F$ is :

Assume $g=10{\text{ms}}^{-2}$

1. $5000\sqrt{\frac{5}{2}}\text{N}$
2. $\frac{5000}{\sqrt{2}}\text{N}$
3. $\frac{10000}{\sqrt{2}}\text{N}$
4. $5000\sqrt{\frac{2}{5}}\text{N}$

Q. Two strings are tied $6\text{m}$ apart on a straight vertical rod and on the other end, a mass of $100\text{g}$ is tied as shown in figure.


Each string is $5\text{m}$ long. Tensions $T_1$ and $T_2$ are experienced by the string when the rod is rotated with $60\text{rpm}$.
Take $g=10{\text{m/s}}^2$ and ${\pi }^2\simeq 10$ for calculation.

1. Value of $T_1$ is $10.83\text{N}$
2. Value of $T_2$ is $9.17\text{N}$
3. Angular velocity of rod is $2\pi \text{rad/s}$
4. Angular velocity of rod is $\pi \text{rad/s}$

Q. A body of mass, $m={10}^{-2}\text{kg}$ is moving in a medium experiencing a frictional force, $F=-k{v}^2$. Its intial speed is, $v_0=20\text{m/s}$. If, after $10\text{seconds}$, its energy is $\frac{1}{8}m{v}_0^2$, then the value of $k$ will be

1. $5\times {10}^{-3}\text{kg/s}$
2. $5\times {10}^{-5}\text{kg/s}$
3. ${10}^{-4}\text{kg/s}$
4. $0.5\times {10}^{-3}\text{kg/s}$

Q. An object of $60\text{kg-wt}$ is weighed by hanging it to the spring balance, from the ceiling of a car. If the car starts moving around a curve of radius $10\text{m}$ at a speed $10{\text{ms}}^{-1}$. The reading of the spring balance would be
[$1\text{kg-wt}=10\text{N}$]

1. $60\sqrt{2}\text{kg-wt}$
2. $600\text{kg-wt}$
3. $120\sqrt{2}\text{kg-wt}$
4. $60\text{kg-wt}$

Q. Forces with magnitudes proportional to $AB,BC$ and $2CA$ act along the sides of a triangle $ABC$ in order. Their resultant is represented in magnitude and direction by:

1. $CA$
2. $AC$
3. $BC$
4. $CB$

Q. Th figure shows a block of mass $m$ moving without friction along three tracks with same speed $v$.

Choose the correct relation

1. $N_1>N_2$
2. $N_2>N_3$
3. $N_3>N_1$
4. $N_1=N_2=N_3$

Q. In a uniform circular motion, the formula $a={\omega }^{2}r$ gives magnitude of acceleration.

1. tangential
2. centripetal
3. no

Q. In a uniform circular motion, the formula $a={\omega }^{2}r$ gives magnitude of acceleration.

1. tangential
2. centripetal
3. no

Q. A body of mass $m$ dropped from a certain height strikes a light vertical fixed spring of stiffness k. The height of its fall before touching the spring if the maximum compression of the spring is equal to $\frac{3mg}{k}$ is

1. $\frac{3mg}{2k}$
2. $\frac{2mg}{k}$
3. $\frac{3mg}{4k}$
4. $\frac{mg}{4k}$

Q. A spring balance is attached to the ceiling of a lift. A man hangs his bag on the spring and the spring balance reads 49 N, when the lift is stationary. Determine the reading of the spring balance, if the lift moves downward with an acceleration of 5 $m{s}^{-2}$.

Q. The figure shows a system of two blocks of $10kg$ and $5kg$ mass, connected by ideal strings and pulleys. Here the ground is smooth and the coefficient of friction between the two blocks is$\mu =0.5$. A horizontal force $F$ is applied on a lower block as shown. The minimum value of $F$ required to start sliding between the blocks is : (Take $g=10m/s^2)$

1. $12.5N$
2. $25N$
3. $50N$
4. $100N$

Q.

What is centripetal acceleration? Give its formula.

Q. The radius of curvature of a railway line at a place when the train is moving with a speed of $36kmh{r}^{-1}$ is $1000m$, the distance between the two rails being $1.5m$. The elevation of the outer rail above the inner rail is:

1. $\frac{1.5}{98}$
2. $\frac{3}{98}$
3. $\frac{4.5}{98}$
4. $\frac{1}{98}$

Q.

A large blade of a helicopter is rotating in a horizontal circle with constant angular velocity. Determine the ratio of acceleration of two particles A and B which are located at a distance of $3\text{cm}$ and $6\text{cm}$ respectively from the center of the circle.

1. $1:1$

2. $1:2$

3. $1:3$

4. $1:6$

Q. A man of mass $60kg$ standing on a light weighting machine kept ina box of mass $30kg.$ 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 manages to keep the box at rest $\left(1\right)$ what is weight shown by the machine ? $\left(2\right)$ what force should the man exert on the rope to get his correct weight on the machine ?

1. 150 N
2. 300 N
3. 600 N
4. 900 N

Q. The potential energy of a conservative system is given by $U=a{y}^2-by$, where $y$ represents the position of the particle, both $a$ and $b$ are constants. What is the force acting on the system?

1. $-by$
2. $-ay$
3. $2ay-b$
4. $b-2ay$