Physics-
General
Easy
Question
Express 0K on Fahrenheit scale.
.
The correct answer is: ![negative 459.67 to the power of ring operator end exponent F](data:image/png;base64,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)
Related Questions to study
physics-
A uniform ladder of length 5 m is placed against the wall as shown in the figure. If coefficient of friction
is the same for both the walls, what is the minimum value of m for it not to slip?
![](data:image/png;base64,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)
A uniform ladder of length 5 m is placed against the wall as shown in the figure. If coefficient of friction
is the same for both the walls, what is the minimum value of m for it not to slip?
![](data:image/png;base64,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)
physics-General
physics-
Two light vertical springs with equal natural lengths and spring constants
and
are separated by a distance l. Their upper ends are fixed to the ceiling and their lower ends to the ends A and B of a light horizontal rod AB. A vertical downward force F is applied at point C on the rod. AB will remain horizontal in equilibrium if the distance AC is
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)
Two light vertical springs with equal natural lengths and spring constants
and
are separated by a distance l. Their upper ends are fixed to the ceiling and their lower ends to the ends A and B of a light horizontal rod AB. A vertical downward force F is applied at point C on the rod. AB will remain horizontal in equilibrium if the distance AC is
![](data:image/png;base64,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)
physics-General
Physics-
The spool shown in fig is placed on a rough horizontal surface and has inner radius
and outer radius R. The angle
between the applied force and the horizontal can be varied. The critical angle
for which the spool does not roll and remains stationary is given by:
![](data:image/png;base64,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)
The spool shown in fig is placed on a rough horizontal surface and has inner radius
and outer radius R. The angle
between the applied force and the horizontal can be varied. The critical angle
for which the spool does not roll and remains stationary is given by:
![](data:image/png;base64,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)
Physics-General
physics-
Find minimum height of the obstacle so that the sphere can stay in equilibrium.
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)
Find minimum height of the obstacle so that the sphere can stay in equilibrium.
![](data:image/png;base64,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)
physics-General
physics-
Two halves of a round homogeneous cylinder are held together by a thread wrapped round the cylinder with two equal weights. The complete cylinder weighs 31.4 kg. The plane of contact of both of its halves is vertical. For equilibrium of both haves of the cylinder, the minimum value of m is
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)
Two halves of a round homogeneous cylinder are held together by a thread wrapped round the cylinder with two equal weights. The complete cylinder weighs 31.4 kg. The plane of contact of both of its halves is vertical. For equilibrium of both haves of the cylinder, the minimum value of m is
![](data:image/png;base64,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)
physics-General
physics-
An unstretched massless rope is wrapped around a solid cylinder of mass
and radius
of the rope is fixed while cylinder is released from rest. The acceleration of the rope is
![](data:image/png;base64,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)
An unstretched massless rope is wrapped around a solid cylinder of mass
and radius
of the rope is fixed while cylinder is released from rest. The acceleration of the rope is
![](data:image/png;base64,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)
physics-General
physics-
A hollow sphere is released from the top of a wedge as shown in the figure. There is no friction between the wedge and the ground. There is sufficient friction to provide pure rolling of the sphere. The velocity of the sphere with respect to ground just before it leaves the wedge will be [Assume masses of the wedge and sphere are equal, h>>R the radius of sphere and the portion of wedge where the sphere leaves the wedge is horizontal]
![](data:image/png;base64,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)
A hollow sphere is released from the top of a wedge as shown in the figure. There is no friction between the wedge and the ground. There is sufficient friction to provide pure rolling of the sphere. The velocity of the sphere with respect to ground just before it leaves the wedge will be [Assume masses of the wedge and sphere are equal, h>>R the radius of sphere and the portion of wedge where the sphere leaves the wedge is horizontal]
![](data:image/png;base64,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)
physics-General
Physics-
A non homogenous solid cylinder of radius r is placed on a rough inclined plane of angle of inclination 60° The angle of repose corresponds to 30°. The maximum radius of gyration in order to prevent slipping is :
A non homogenous solid cylinder of radius r is placed on a rough inclined plane of angle of inclination 60° The angle of repose corresponds to 30°. The maximum radius of gyration in order to prevent slipping is :
Physics-General
physics-
Four solid spheres P, Q, R and S are made to move on a horizontal surface which is rough enough. Sphere P is given a spin and released sphere Q is given a linear velocity and released ; sphere R is given a reverse spin and a forward linear velocity and released ; sphere S is given a forward spin and a forward linear velocity as shown in figure the direction of frictional force F for P, Q, R and S will be along
![](data:image/png;base64,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)
Four solid spheres P, Q, R and S are made to move on a horizontal surface which is rough enough. Sphere P is given a spin and released sphere Q is given a linear velocity and released ; sphere R is given a reverse spin and a forward linear velocity and released ; sphere S is given a forward spin and a forward linear velocity as shown in figure the direction of frictional force F for P, Q, R and S will be along
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)
physics-General
physics-
A disc of radius 20 cm is rolling with slipping on a flat horizontal surface. At a certain instant the velocity of its centre is 4 m/s and its angular velocity is 10rad/s. The lowest contact point is O
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)
A disc of radius 20 cm is rolling with slipping on a flat horizontal surface. At a certain instant the velocity of its centre is 4 m/s and its angular velocity is 10rad/s. The lowest contact point is O
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)
physics-General
physics-
A hollow cylinder with inner radius R, outer radius 2R and mass M is rolling with speed of its axis v. Its kinetic energy is
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)
A hollow cylinder with inner radius R, outer radius 2R and mass M is rolling with speed of its axis v. Its kinetic energy is
![](data:image/png;base64,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)
physics-General
physics-
A disc of radius R rolls without slipping at speed
along positive x-axis. Velocity of point P at the instant shown in fig is
![](data:image/png;base64,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)
A disc of radius R rolls without slipping at speed
along positive x-axis. Velocity of point P at the instant shown in fig is
![](data:image/png;base64,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)
physics-General
physics-
Fig shows smooth inclined plane of inclination
fixed in a car. A sphere is set in pure rolling on the incline. For what value of a the acceleration of car in horizontal direction) the sphere will continue pure rolling.
![](data:image/png;base64,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)
Fig shows smooth inclined plane of inclination
fixed in a car. A sphere is set in pure rolling on the incline. For what value of a the acceleration of car in horizontal direction) the sphere will continue pure rolling.
![](data:image/png;base64,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)
physics-General
physics-
The arrangement shown in figure consists of two identical uniform solid cylinders, each of mass m, on which two light threads are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axile of the upper cylinder is assumed to be absent.
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)
The arrangement shown in figure consists of two identical uniform solid cylinders, each of mass m, on which two light threads are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axile of the upper cylinder is assumed to be absent.
![](data:image/png;base64,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)
physics-General
physics-
A uniform cylinder of mass M and radius R rolls without slipping down a slope of angle
with horizontal. The cylinder is connected to a spring of force constant k at the centre, the other side of which is connected to a fixed support at A. The cylinder is released when the spring is unstretched. The force of friction f)
![](data:image/png;base64,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)
A uniform cylinder of mass M and radius R rolls without slipping down a slope of angle
with horizontal. The cylinder is connected to a spring of force constant k at the centre, the other side of which is connected to a fixed support at A. The cylinder is released when the spring is unstretched. The force of friction f)
![](data:image/png;base64,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)
physics-General