Advertisements
Advertisements
प्रश्न
A block of mass m is kept on a horizontal ruler. The friction coefficient between the ruler and the block is μ. The ruler is fixed at one end and the block is at a distance L from the fixed end. The ruler is rotated about the fixed end in the horizontal plane through the fixed end. (a) What can the maximum angular speed be for which the block does not slip? (b) If the angular speed of the ruler is uniformly increased from zero at an angular acceleration α, at what angular speed will the block slip?
उत्तर
(a) Given:
Mass of the block = m
Friction coefficient between the ruler and the block = μ
Let the maximum angular speed be ω1 for which the block does not slip
Now, for the uniform circular motion in the horizontal plane, we have :
\[\mu \text{mg = m }\omega_1^2 L\]
\[ \therefore \omega_1 = \sqrt{\frac{\mu g}{L}}\]
(b) Let the block slip at an angular speed ω2.
For the uniformly accelerated circular motion, we have :
\[\mu \text{mg} = \sqrt{\left( \text{ m }\omega_2^2 L \right)^2 + \left( \text{mL } \alpha^2 \right)^2}\]
\[ \Rightarrow \omega_2^4 + \alpha^2 = \frac{\mu^2 g^2}{L^2}\]
\[\mu \text{mg } = \sqrt{\left( \text{ m }\omega_2^2 L \right)^2 + \left( \text{ mL } \alpha^2 \right)^2}\]
\[ \Rightarrow \omega_2^4 + \alpha^2 = \frac{\mu^2 g^2}{L^2}\]
\[\Rightarrow \omega_2 = \left[ \left( \frac{\mu g}{L} \right)^2 - \alpha^2 \right]^{1/4}\]
APPEARS IN
संबंधित प्रश्न
A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with a speed of 40 rev/min in a horizontal plane. What is the tension in the string? What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N?
You may have seen in a circus a motorcyclist driving in vertical loops inside a ‘death-well’ (a hollow spherical chamber with holes, so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 25 m?
A smooth block loosely fits in a circular tube placed on a horizontal surface. The block moves in a uniform circular motion along the tube. Which wall (inner or outer) will exert a nonzero normal contact force on the block?
Tow cars having masses m1 and m2 moves in circles of radii r1 and r2 respectively. If they complete the circle in equal time, the ratio of their angular speed ω1/ω2 is
A car moves at a constant speed on a road as shown in figure. The normal force by the road on the car NA and NB when it is at the points A and B respectively.
A motorcycle is going on an overbridge of radius R. The driver maintains a constant speed. As the motorcycle is ascending on the overbridge, the normal force on it
A train A runs from east to west and another train B of the same mass runs from west to east at the same speed along the equator. A presses the track with a force F1 and B presses the track with a force F2.
If the earth stop rotating, the apparent value of g on its surface will
Let θ denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of the bob is m, the tension is the string is mg cos θ
A particle is going in a spiral path as shown in figure with constant speed.
A mosquito is sitting on an L.P. record disc rotating on a turn table at \[33\frac{1}{3}\] revolutions per minute. The distance of the mosquito from the centre of the turn table is 10 cm. Show that the friction coefficient between the record and the mosquito is greater than π2/81. Take g =10 m/s2.
A simple pendulum is suspended from the ceiling of a car taking a turn of radius 10 m at a speed of 36 km/h. Find the angle made by he string of the pendulum with the vertical if this angle does not change during the turn. Take g = 10 m/s2.
Suppose the amplitude of a simple pendulum having a bob of mass m is θ0. Find the tension in the string when the bob is at its extreme position.
Choose the correct option.
Select correct statement about the formula (expression) of moment of inertia (M.I.) in terms of mass M of the object and some of its distance parameter/s, such as R, L, etc.
A body slides down a smooth inclined plane having angle θ and reaches the bottom with velocity v. If a body is a sphere, then its linear velocity at the bottom of the plane is
A child starts running from rest along a circular track of radius r with constant tangential acceleration a. After time the feels that slipping of shoes on the ground has started. The coefficient of friction between shoes and the ground is [g = acceleration due to gravity].
A racing car travels on a track (without banking) ABCDEFA (Figure). ABC is a circular arc of radius 2 R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is µ = 0.1. The maximum speed of the car is 50 ms–1. Find the minimum time for completing one round.
A racing car is travelling along a track at a constant speed of 40 m/s. A T.V. cameraman is recording the event from a distance of 30 m directly away from the track as shown in the figure. In order to keep the car under view in the position shown, the angular speed with which the camera should be rotated is ______.
A stone tide to a string of length L is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of change in its velocity, as it reaches a position where the string is horizontal, is `sqrt(x("u"^2 - "gL")`. The value of x is ______.
Define centripetal force.
