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## Newton second law and pendulum

A pendulum is hanged in the car.
Car starts to move by constant acceleration
How to measure acceleration by angle the pendulum?

If the pendulum makes an angle θ with the vertical, then a = g tan θ

This is based on the idea of pseudo force experienced by a body in an accelerated frame of reference.

When a body is in an accelerated frame of reference, it feels as if it is acted upon by a force equal to its mass multiplied by the acceleration of the system. Please have a look at the diagram below.

## Numerical Problems from “Uniform Circular Motion” –Online Home work for KV Pattom Class 11 students

All students of class XI are to copy down the questions and solve it in the home work copy and solve it to submit on or before 22 Sept 2014.

1. What is the angular velocity of a second hand and minute hand of a clock?
2. A body of mass 0.4 kg is whirled in a horizontal circle of radius 2 m with a constant speed of 10 ms . Calculate its (i) angular speed (ii) frequency of revolution (iii) time period and (iv) centripetal acceleration.
3. A circular wheel of 0.50 m radius is moving with a speed of10 ms. Find the angular speed.
4. Assuming that the moon completes one revolution in a circular orbit around the earth in 27.3 days, calculate the acceleration of the moon towards the earth. The radius of the circular orbit can betaken as 3.85 x 10 km.
5. The angular velocity of a particle moving along a circle of radius 560 cm is increased in 5 minutes from 100 revolutions per minute to 400 revolutions per minute. Find angular acceleration and (ii) linear acceleration.
6. Calculate the linear acceleration of a particle moving in a circle of radius 0.4 m at the instant when its angular velocity is 2 rad /s and its angular acceleration is 5 rad/s2
7. A threaded rod with 12 turns per cm and diameter 1.18 cm is mounted horizontally. A bar with a threaded hole to match the rod is screwed  onto the rod. The bar spins at the rate of 216 rpm. How long will it take for the bar to move 1.50 cm along the rod.

## Projectile Motion – Numerical Problems (Assignment for KV Pattom Class XI students)

The students are requested to start solving these numerical problems. More questions will be added till 8 pm on 13/09/2014. There will be more than 20 questions in all. All students are expected to solve at least 20 numericals among these. The deadline form submission is 15/09/2014.

1. A football player kick a ball at an angle of 37° to the horizontal with an initial speed of 15 m/s. Assuming that the ball travels in a vertical plane, calculate (i) the time at which the ball reaches the highest point. (ii) the maximum height reached by the ball (iii) the horizontal range of the projectile and (iv) the time for which the ball is in air.
2. A body is projected with a velocity of 20 m/s in a direction making an angle 60° with the horizontal. CCalculate (i) position after 0.5 seconds and  (ii) velocity after 0.5 seconds
3. The maximum vertical height of a projectile is 10 m. If the magnitude of the initial velocity is 28 m/s, what is the direction of the initial velocity? (g=9.8 m/s)
4. A bullet fired from a gun with a velocity of 140 m/s strikes the ground at the same level as the gun at a distance of 1 km. Find the angle of inclination with the horizontal at which the bullet is fired. (g=9.8 m/s)
5. A bullet is fired at an angle of 15° with the horizontal and hits the ground 6 km away. Is it possible to hit a target 10 km away by adjusting the angle of projection assuming the initial speed to be the same?
6. A cricketer can throw a ball to a maximum horizontal distance of 160 m. Calculate the maximum vertical height to which he can throw the ball. (g=10  m/s)
7. A football is kicked with 20 m/s at a projection angle of 45°. A receiver on the goal line 25 metres away in the direction of the kick runs the same instant to meet the ball. What must be his speed, if he is to catch the ball before it hits the ground?
8. A bullet fired at an angle of 60° with the vertical hits the ground at a distance of 2 km. Calculate the distance at which the bullet will hit the ground when fired at an angle of 45°, assuming the speed to be the same.
9. A person observes a bird on tree 39.6 m high and at a distance of 59.2m. With what velocity the person should shoot an arrow at an angle of 45° so that it may hit the bird?
10. A ball is thrown from the top of a tower with an initial velocity of 10 m/s at an angle of 30° with the horizontal. If it hits the ground at a distance of 17.3 m from the base of the tower, calculate the height of the tower. (Given g =10  m/s)
11. Prove that the time of flight T and the horizontal range R of a projectile are connected by the equation gT2 =2R tanθ, where θ is the angle of projection.
12. Show that the range of a projectile for two angles α and β is same if α+β=90°
13. A body is projected with velocity of 40 m/s. After 2 s it crosses a vertical pole of height 20.4 m. Calculate the angle of projection and horizontal range.
14. A plane is flying horizontally at a height of 1000 m with a velocity of 100 m/s when a bomb is released from it. Find (i) the time taken by it to reach the ground. (ii) the velocity with which the bomb hits the target and (iii) the distance of the target.
15. From the top of a building 19.6 m high a ball is projected horizontally. After how long does it strike the ground? If the line joining the point of projection to the point where it hits the ground makes an angle of 45° with the horizontal, what is the initial velocity of the ball.
16. A body is thrown horizontally from the top of a tower and strikes the ground after 2 seconds at an angle of 45° with the horizontal. Find the height of the tower and the speed with which the body was thrown. Take g =9.8 m/s)
17. A ball is projected horizontally from a tower with a velocity of 4 m/s. Find the velocity of the ball after 0.7 s. (Given g =10  m/s)
18. In between two hills of heights 100 m and 92 m respectively , there is a valley of breadth 16m. If a vehicle jumps from the first hill to the second, what must be the minimum horizontal velocity so that it may not fall into the valley? (Given g =10  m/s)
19. A mailbag is to be dropped into a post office from an aeroplane flying horizontally with a velocity of 270 km/h at a height of 176.4 m above the ground. How far must the aeroplane be from the post office at the time of dropping the bag so that the bag directly falls into the post office?
20. An aeroplane is flying in a horizontal direction with a velocity of 600 km/h and at a height of 1960 m. When it is above a point A on ground an object is dropped from it. The object strike the ground at the point B. Find the distance AB.
21. Two tall buildings are situated 200 m apart. With what speed must a ball be thrown horizontally from the window 540 m above the ground in one building so that it will enter a window 50 m above the ground in the other?

## Kinematics Problem

Three particles A,B,C are situated at vertices of a equilateral triangle ABC of side ‘a’ m at t=0 Each of the particle moves with constant speed ‘ v ‘. A always has it’s velocity along AB ,B always has it’s velocity along BC , C always has it’s velocity along CA. Derive the equation of trajectory of any one particle (means find y=f(x) i.e. relation between y and x coordinates of the particle). Also find the rate of rotation of the triangle formed by joining the lines connecting the three points as a function of time. If you think it requires more information such as acceleration etc., introduce them if needed.

## Kinematics Problem

Three particles A,B,C are situated at vertices of a equilateral triangle ABC of side ‘a’ m at t=0 Each of the particle moves with constant speed ‘ v ‘. A always has it’s velocity along AB ,B always has it’s velocity along BC , C always has it’s velocity along CA. Derive the equation of trajectory of any one particle (means find y=f(x) i.e. relation between y and x coordinates of the particle). Also find the rate of rotation of the triangle formed by joining the lines connecting the three points as a function of time. If you think it requires more information such as acceleration etc., introduce them if needed.

## Four persons K, L , M, N are initially at the four corners of a square of side d Problem

Four persons K, L , M, N are initially at the four corners of a square of side ‘d’. Each person now starts moving with a uniform speed ‘v’ in such a way that K always moves directly towards L, L towards M, M towards N and N towards K. After what time will they meet?

Answer: Many students do not understand the real situation initially. Every time the persons are approaching each other and hence they will be moving closer and closer as they continue their walking. Finally they’ll reach the centre of the square to meet each other. I’ve tried to visualize the situation below.

From the diagram, we can make out that the resultant displacement by each when they meet will be d/√2 and the component of velocity of each towards the final point (the centre of the square) is v/√2.

Therefore,the time taken = displacement by the component of velocity in its direction = (d/√2) / (v/√2) = d/v

## A thought provoking (HOTS) problem from rotation

Consider a disk rotating in a horizontal plane with a constant angular speed “omega” about its center O.

The disk has a shaded region on one side of the diameter and an unshaded region on the other  side.

When disk is in horizontal plane x-y with left half shaded and right half unshaded, two pebbles P and Q are simultaneously projected at an angle towards R.

The velocity of projection is in the y-z plane and is same for both pebbles with respect to the disk.

Assume that

(i) they land back on the disk before the disk has completed one-eighth rotation,

(ii) their range is less than half the disk  radius,and

(iii)”omega” remains constant throughout.

Then where will the two pebbles land?

(All are invited to answer this)

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