Advertisements
Advertisements
प्रश्न
A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle θ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is μ. Find the range of the angular speed for which the block will not slip.
Advertisements
उत्तर
When the bowl rotates at maximum angular speed, the block tends to slip upwards.
Also, the frictional force acts downward.
Here, we have:
Radius of the path that the block follow = r = Rsinθ
Let N1 be the normal reaction on the block and ω1 be the angular velocity after which the block will slip.
From the free body diagram-1, we get :
\[N_1 - \text{mg}\cos\theta = {m\omega}_1^2 (R\sin\theta)\sin\theta . . . \](i)
\[\mu N_1 + \text{mg}sin\theta = m \omega_1^2 (R\sin\theta)\cos\theta . . . \](ii)
On solving the two equation, we get :
\[\omega_1 = \left[ \frac{g(\text{sin}\theta + \mu\text{ cos}\theta)}{\text{R sin}\theta (\text{cos}\theta - \mu \text{ sin }\theta)} \right]^{1/2}\]
Let us now find the minimum speed (ω2) on altering the direction of \[\mu\] (as shown in figure):
\[\omega_2 = \left[ \frac{g(\text{ sin }\theta - \mu \text{ cos }\theta)}{\text{R sin}\theta (\text{ cos }\theta + \mu \text{ sin }\theta)} \right]^{1/2}\]
Hence, the range of speed is between ω2 and ω1.
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?
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?
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
If the earth stop rotating, the apparent value of g on its surface will
A rod of length L is pivoted at one end and is rotated with a uniform angular velocity in a horizontal plane. Let T1 and T2 be the tensions at the points L/4 and 3L/4 away from the pivoted ends.
A stone is fastened to one end of a string and is whirled in a vertical circle of radius R. Find the minimum speed the stone can have at the highest point of the circle.
The bob of a simple pendulum of length 1 m has mass 100 g and a speed of 1.4 m/s at the lowest point in its path. Find the tension in the string at this instant.
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?
In a children's park a heavy rod is pivoted at the centre and is made to rotate about the pivot so that the rod always remains horizontal. Two kids hold the rod near the ends and thus rotate with the rod (In the following figure). Let the mass of each kid be 15 kg, the distance between the points of the rod where the two kids hold it be 3.0 m and suppose that the rod rotates at the rate of 20 revolutions per minute. Find the force of friction exerted by the rod on one of the kids.
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.
In non-uniform circular motion, the ratio of tangential to radial acceleration is (r = radius, a = angular acceleration and v = linear velocity)
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].
The real force 'F' acting on a particle of mass ' m' performing circular motion acts along the radius of circle 'r' and is directed towards the centre of circle. The square root of the magnitude of such force is (T = periodic time).
Angular displacement (θ) of a flywheel varies with time as θ = at + bt2 + ct3 then angular acceleration is given by ____________.
A body is moving along a circular track of radius 100 m with velocity 20 m/s. Its tangential acceleration is 3 m/s2 then its resultant accelaration will be ______.
A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius 20 cm. If the block takes 40 s to complete one round, the normal force by the side walls of the groove 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 ______.
Statement I: A cyclist is moving on an unbanked road with a speed of 7 kmh-1 and takes a sharp circular turn along a path of radius of 2 m without reducing the speed. The static friction coefficient is 0.2. The cyclist will not slip and pass the curve. (g = 9.8 m/s2)
Statement II: If the road is banked at an angle of 45°, cyclist can cross the curve of 2 m radius with the speed of 18.5 kmh-1 without slipping.
In the light of the above statements, choose the correct answer from the options given below.
