Wedge Problem

Q. A sphere rolls down on an inclined plane of inclination $\theta$. What is the acceleration as the sphere reaches bottom

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Q. Why does a cyclist bend or lean during a turn?

Q. A ring, solid sphere, and a disc are rolling down from the top of the same height, then the sequence to reach on surface is?

1. Ring, disc, sphere
2. Disc, ring, sphere

3. Sphere, ring, disc

4. Sphere, disc, ring

Q. A solid sphere and a disc of same mass and radius starts rolling down a rough inclined plane, from the same height the ratio of the time taken in the two cases is

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Q. A solid sphere of mass 0.1 kgand radius 2 cmrolls down an inclined plane 1.4min length at an angle whose slope is 1/10. If sphere started from rest, its final velocity will be

1. 0.14 m/sec
2. 0.7 m/sec
3. 14 m/sec
4. 1.4 m/sec

Q. A sphere rolls down on an inclined plane of inclination $\theta$. What is the acceleration as the sphere reaches bottom

1. $\frac{5}{7}g\sin \theta$
2. $\frac{3}{5}g\sin \theta$
3. $\frac{2}{7}g\sin \theta$
4. $\frac{2}{5}g\sin \theta$

Q.

A cylinder is released from rest from the top of an inclined plane of inclination $\theta$ and length l. If the cylinder rolls without slipping, what will be its speed when it reaches the bottom?

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Q. A solid cylinder of mass 'M' and radius 'R' rolls without slipping down an inclined plane of length 'L' and height 'h'. What is the speed of its centre of mass when the cylinder reaches its bottom

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4. 

Q.

A solid cylinder rolls down a fixed inclined plane of height h and inclination $\theta$. Calculate the time taken to reach bottom?

{Assume the mass be M and radius be R}

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Q. A long horizontal rod has a bead which can slide along its length, the bead is initially at a distance L from the end of the rod. The rod is set in angular motion in the horizontal plane about end A with constant angular acceleration . If the coefficient of the spring is u, and the gravity is neglected then plot the graph between frictional force acting on bead and the time.

Q. A solid sphere rolls purely down an inclined plane such that it gains angular speed with time. What is the direction of frictional force on the sphere?

1. Upwards along the incline
2. zero
3. Sometimes upwards and sometimes downwards.
4. Downwards along the incline

Q. A solid sphere rolls purely down an inclined plane such that it gains angular speed with time. What is the direction of frictional force on the sphere?

1. Downwards along the incline
2. Upwards along the incline
3. zero
4. Sometimes upwards and sometimes downwards.

Q. During rotation of a body, the position vector is along X axis and force vector is along Y axis. The direction of torque vector is

1. in the X-Y plane
2. Along negative Z-axis
3. Along positive Z-axis
4. in the X-Z plane

Q. Initial angular velocity of a circular disc of mass $M$ is ${\omega }_1$. Then two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?

1. $\left(\frac{M+m}{M}\right){\omega }_1$
2. $\left(\frac{M+m}{m}\right){\omega }_1$
3. $\left(\frac{M}{M+4m}\right){\omega }_1$
4. $\left(\frac{M}{M+2m}\right){\omega }_1$

Q. A solid sphere of mass $M$ and radius $2R$ rolls down an inclined plane of height $h$ without slipping. The speed of its center of mass when it reaches the bottom is

1. $\sqrt{\frac{6}{7}gh}$
2. $\sqrt{3gh}$
3. $\sqrt{\frac{4}{3}gh}$
4. $\sqrt{\frac{10}{7}gh}$

Q. A ring of radius $r$ made of wire of density $\rho$ is rotated about a stationary vertical axis passing through its centre and perpendicular to the plane of the ring as shown in the figure. Determine the angular velocity (in rad/s) of ring at which the ring breaks. The wire breaks at tensile stress $\sigma$. Ignore gravity. Take $\sigma /\rho =4$ and $r=1m$.

Q. A uniform thin rod of mass $M$, length $L$ is vertically straight on a horizontal floor. If the rod is falling to a side without slipping at its bottom, the moment of inertia is:

1. $\frac{M{L}^2}{3}$
2. $\frac{M{L}^2}{12}$
3. $\frac{M{L}^2}{9}$
4. $\frac{M{L}^2}{6}$

Q. A uniform disc of radius $R$ and mass $M$ is free to rotate only about its axis. A string wrapped over its rim and a body of mass $m$ is tied to the free end of the string as shown in the figure. Then the acceleration of the body is:

Q.

A particle P is sliding down a frictionless hemispherical bowl. It passes the point A at t = 0. At this instant of time, the horizontal component of its velocity is v. A bead Q of the same mass as P is ejected from A at t = 0 along the horizontal string AB, with a speed v. Friction between the bead and the string may be neglected. Let tp and tq be the respective times taken by P and Q to reach the point B. Then

Time taken by p is

a) less than Q

b) greater than Q

c)equal to Q

Q. A solid sphere (mass $2M$ ) and a thin hollow spherical shell (mass $M$) both of the same size, roll down an inclined plane, then

1. Solid sphere will reach the bottom first
2. Hollow spherical shell will reach the bottom first
3. Both will reach at the same time
4. None of these

Q. A solid sphere rolling on a rough horizontal surface with a linear speed $\nu$ collides elastically with a fixed, smooth vertical wall. Find the speed of the sphere after it has began pure rolling in the backwards direction.

1. $\frac{4}{7}\nu$
2. $\frac{2}{7}\nu$
3. $\frac{3}{7}\nu$
4. None

Q. A ring has mass $M$, radius $R$. A point mass $m$ is placed at a distance $x$ on the axial line as shown in the figure. Find $x$, so that force experienced is maximum

1. $\frac{R}{\sqrt{2}}$
2. $\frac{R}{3\epsilon }$
3. $\frac{R}{2}$
4. $\frac{R}{\sqrt{3}}$

Q. Two torsion pendula (P, Q) are shown. They differ only by the addition of cylindrical masses. The radius of each additional mass is $\frac{1}{4}$th of the radius of the disc. If each cylinder and disc have equal mass, then

1. moment of inertia of Q is $\frac{16}{27}$MR${}^2$
2. moment of inertia of Q is $\frac{27}{16}$MR${}^2$
3. ratio of periods of P and Q is ${\left(\frac{2}{3}\right)}^{\frac{3}{2}}$
4. ratio of periods of P and Q is ${\left(\frac{3}{4}\right)}^{\frac{1}{2}}$

Q.

A cylinder is released from rest from the top of an inclined plane of inclination $\theta$ and length 'l'. If the cylinder rolls without slipping, what will be its speed when it reaches the bottom?

1. $\sqrt{\frac{3}{4}glcos\theta }$

2. $\sqrt{\frac{3}{4}glsin\theta }$

3. $\sqrt{\frac{3}{4}gltan\theta }$

4. $\sqrt{\frac{3}{4}glcot\theta }$

Q. A solid sphere rolls purely down an inclined plane such that it gains angular speed with time. What is the direction of frictional force on the sphere?

1. Downwards along the incline
2. Upwards along the incline
3. zero
4. Sometimes upwards and sometimes downwards.

Q. The radius of gyration of a circular disc of mass $m$ rotating about an axis perpendicular to its plane and passing through the center is:

1. $2R$
2. $1.4R$
3. $R/1.4$
4. $R/2$

Q. A wedge of mass $M$ lies on a smooth fixed wedge of inclination $\theta$. A particle of mass $m$ lies on the wedge of mass $M$ in the figure. All surfaces are frictionless and the distance of the block from the fixed wedge is $\ell$. Find.
(a) Normal reaction between $m$ and $M$
(b) the time after which the particle will hit the fixed plane.

Q. The temperature gradient in a rod of $0.5m$ long is ${80}^{o}C/m$. If the temperature of hotter end of the rod is ${30}^{o}C$, then the temperature of the cooler end is

1. $0^{o}C$
2. ${-10}^{o}C$
3. ${40}^{o}C$
4. ${10}^{o}C$

Q. A uniform cube of mass M is floating on the surface of a liquid with three fourth of its volume immersed in the liquid(density $=\rho$). The length of the side of the cube is equal to?

Q. A square plate of edge ${}^{\prime }{a}^{\prime }$ and a circular disc of same diameter are placed touching each other at the midpoint of an edge of the plate as shown in figure. If mass per unit area for the two plates are same then find the distance of center of mass of the system from the centre of the disc.