Advertisements
Advertisements
Question
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.
Options
T1 > T2
T2 > T1
T1 = T2
The relation between T1 and T2 depends on whether the rod rotates clockwise or anticlockwise.
Advertisements
Solution
T1 > T2
Let the angular velocity of the rod be \[\omega\] .
Distance of the centre of mass of portion of the rod on the right side of L/4 from the pivoted end :
\[r_1 = \frac{L}{4} + \frac{1}{2}\left( \frac{3L}{4} \right) = \frac{5L}{8}\]
Mass of the rod on the right side of L/4 from the pivoted end : \[\text{m}_1 = \frac{3}{4}\text{M}\]
At point L/4, we have :
\[T_1 = \text{ m}_1 \omega^2 \text{ r}_1 \]
\[ = \frac{3}{4}\text{ M } \omega^2 \frac{5}{8}\text{ L} = \frac{15}{32}\text{ M }\omega^2 \text{ L}\]
Distance of the centre of mass of rod on the right side of 3L/4 from the pivoted end :
\[\text{r}_1 = \frac{1}{2}\left( \frac{L}{4} \right) + \frac{3L}{4} = \frac{7L}{8}\]
Mass of the rod on the right side of L/4 from the pivoted end : \[\text{m}_1 = \frac{1}{4}\text{M}\]
At point 3L/4, we have :
\[\text{T}_2 = \text{m}_2 \omega^2 \text{r}_2 \]
\[ = \frac{1}{4}\text{M} \omega^2 \frac{7}{8}\text{L} = \frac{7}{32}\text{M} \omega^2 \text{L}\]
∴ T1 > T2
APPEARS IN
RELATED QUESTIONS
A thin circular loop of radius R rotates about its vertical diameter with an angular frequency ω. Show that a small bead on the wire loop remains at its lowermost point for `omega <= sqrt(g/R)` .What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for `omega = sqrt("2g"/R)` ?Neglect friction.
When a particle moves in a circle with a uniform speed
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.
Water in a bucket is whirled in a vertical circle with string attached to it. The water does no fall down even when the bucket is inverted at the top of its path. We conclude that in this position
Three identical cars A, B and C are moving at the same speed on three bridges. The car A goes on a place bridge, B on a bridge convex upward and C goes on a bridge concave upward. Let FA, FB and FC be the normal forces exerted by the car on the bridges when they are at the middle of bridges.
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.
A particle is going in a spiral path as shown in figure with constant speed.
Find the acceleration of a particle placed on the surface of the earth at the equator due to earth's rotation. The diameter of earth = 12800 km and it takes 24 hours for the earth to complete one revolution about its axis.
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 motorcycle has to move with a constant speed on an over bridge which is in the form of a circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point.(a) What can its maximum velocity be for which the contact with the road is not broken at the highest point? (b) If the motorcycle goes at speed 1/√2 times the maximum found in part (a), where will it lose the contact with the road? (c) What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge?
A car goes on a horizontal circular road of radius R, the speed increasing at a constant rate \[\frac{\text{dv}}{\text{dt}} = a\] . The friction coefficient between the road and the tyre is μ. Find the speed at which the car will skid.
A block of mass m moves on a horizontal circle against the wall of a cylindrical room of radius R. The floor of the room on which the block moves is smooth but the friction coefficient between the wall and the block is μ. The block is given an initial speed v0. As a function of the speed v writes
(a) the normal force by the wall on the block,
(b) the frictional force by a wall, and
(c) the tangential acceleration of the block.
(d) Integrate the tangential acceleration \[\left( \frac{dv}{dt} = v\frac{dv}{ds} \right)\] to obtain the speed of the block after one revolution.
A person stands on a spring balance at the equator. 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?
A particle of mass 1 kg, tied to a 1.2 m long string is whirled to perform the vertical circular motion, under gravity. The minimum speed of a particle is 5 m/s. Consider the following statements.
P) Maximum speed must be `5sqrt5` m/s.
Q) Difference between maximum and minimum tensions along the string is 60 N.
Select the correct option.
A particle of mass m is performing UCM along a circle of radius r. The relation between centripetal acceleration a and kinetic energy E is given by
Two particles A and B are located at distances rA and rB respectively from the centre of a rotating disc such that rA > rB. In this case, if angular velocity ω of rotation is constant, then ______
A rope is wound around a solid cylinder of mass 1 kg and radius 0.4 m. What is the angular acceleration of cylinder, if the rope is pulled with a force of 25 N? (Cylinder is rotating about its own axis.)
A particle performs uniform circular motion in a horizontal plane. The radius of the circle is 10 cm. If the centripetal force F is kept constant but the angular velocity is halved, the new radius of the path will be ______.
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 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.
