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Q5.22:
If, 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, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks :
If, 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, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks :
Solution :
Part 1: Calculating Tension in the String
Given Variables:
- Mass of the stone, $m = 0.25\text{ kg}$
- Radius of the circle, $r = 1.5\text{ m}$
- Angular frequency, $\nu = 40\text{ rev/min} = \frac{40}{60}\text{ rev/s} = \frac{2}{3}\text{ rev/s}$
Step 1: Determine Angular Velocity ($\omega$)
The relation between angular frequency and angular velocity is given by $\omega = 2\pi\nu$ [Definition of Angular Velocity].
$\omega = 2 \times \pi \times \frac{2}{3} = \frac{4\pi}{3}\text{ rad/s} \approx 4.189\text{ rad/s}$
Step 2: Calculate Centripetal Force ($T$)
The tension in the string provides the necessary centripetal force for horizontal circular motion [Per Newton's Second Law].
$T = F_c = m\omega^2r$
$T = 0.25 \times \left(\frac{4\pi}{3}\right)^2 \times 1.5$
$T = 0.25 \times \frac{16\pi^2}{9} \times 1.5$
$T = 0.25 \times 1.777 \times 9.87 \times 1.5 \approx 6.58\text{ N}$
Part 2: Calculating Maximum Speed ($v_{max}$)
Given: $T_{max} = 200\text{ N}$
Step 3: Relate Tension to Linear Speed
Using the centripetal force formula $T = \frac{mv^2}{r}$, we solve for $v$ [Centripetal Force Axiom]:
$v_{max} = \sqrt{\frac{T_{max} \times r}{m}}$
$v_{max} = \sqrt{\frac{200 \times 1.5}{0.25}} = \sqrt{\frac{300}{0.25}} = \sqrt{1200}$
$v_{max} \approx 34.64\text{ m/s}$
Part 3: Trajectory Analysis
When the string breaks, the centripetal force vanishes [Per Newton's First Law of Motion]. According to the law of inertia, the stone will continue to move in the direction of its instantaneous velocity at the point of release.
Conclusion: The stone will fly off along the tangent to the circular path at the point where the string breaks.
| Parameter | Calculated Value |
|---|---|
| Tension ($T$) | $6.58\text{ N}$ |
| Max Speed ($v_{max}$) | $34.64\text{ m/s}$ |
Final Solution: The tension in the string is 6.58 N, the maximum speed is 34.64 m/s, and the trajectory of the stone is along the tangent to the circular path at the point of breakage.
More Questions from Class 11 Physics Laws of Motion EXERCISES
- Q4.1(a): Give the magnitude and direction of the net force acting on a drop of rain falling down with a constant speed,
- Q4.1(b): Give the magnitude and direction of the net force acting on a cork of mass 10 g floating on water,
- Q4.1(c): Give the magnitude and direction of the net force acting on a kite skillfully held stationary in the sky,
- Q4.1(d): Give the magnitude and direction of the net force acting on a car moving with a constant velocity of 30 km/h on a rough road,
- Q4.1(e): Give the magnitude and direction of the net force acting on a high-speed electron in space far from all material objects, and free of electric and magnetic fields.
- Q4.10: A body of mass 0.40 kg moving initially with a constant speed of 10 (m s^{-1}) to the north is subject to a constant force of 8.0 N directed towards the south for 30 s. Take the instant the force is applied to be (t = 0), the position of the body at that time to be (x = 0), and predict its position at (t = -5) s, 25 s, 100 s.
- Q4.11(a): A truck starts from rest and accelerates uniformly at 2.0 (m s^{-2}). At (t = 10) s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the velocity of the stone at (t = 11)s ? (Neglect air resistance.)
- Q4.11(b): A truck starts from rest and accelerates uniformly at 2.0 (m s^{-2}). At (t = 10) s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the acceleration of the stone at (t = 11)s ? (Neglect air resistance.)
- Q4.12(a): A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is 1 (m s^{-1}). What is the trajectory of the bob if the string is cut when the bob is at one of its extreme positions,
- Q4.12(b): A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is 1 (m s^{-1}). What is the trajectory of the bob if the string is cut when the bob is at its mean position.
- Q4.13(a): A man of mass 70 kg stands on a weighing scale in a lift which is moving upwards with a uniform speed of 10 (m s^{-1}). What would be the readings on the scale in this case?
- Q4.13(b): A man of mass 70 kg stands on a weighing scale in a lift which is moving downwards with a uniform acceleration of 5 (m s^{-2}). What would be the readings on the scale in this case?
- Q4.13(c): A man of mass 70 kg stands on a weighing scale in a lift which is moving upwards with a uniform acceleration of 5 (m s^{-2}). What would be the readings on the scale in this case?
- Q4.13(d): A man of mass 70 kg stands on a weighing scale in a lift. What would be the reading if the lift mechanism failed and it hurtled down freely under gravity ?
- Q4.14(a): Figure 4.16 shows the position-time graph of a particle of mass 4 kg. What is the force on the particle for (t < 0), (t > 4) s, (0 < t < 4) s? (Consider one-dimensional motion only).
- Q4.14(b): Figure 4.16 shows the position-time graph of a particle of mass 4 kg. What is the impulse at (t = 0) and (t = 4) s ? (Consider one-dimensional motion only).
- Q4.15(i): Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string. A horizontal force (F = 600) N is applied to the 10 kg body (A) along the direction of string. What is the tension in the string?
- Q4.15(ii): Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string. A horizontal force (F = 600) N is applied to the 20 kg body (B) along the direction of string. What is the tension in the string?
- Q4.16: Two masses 8 kg and 12 kg are connected at the two ends of a light inextensible string that goes over a frictionless pulley. Find the acceleration of the masses, and the tension in the string when the masses are released.
- Q4.17: A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.
- Q4.18: Two billiard balls each of mass 0.05 kg moving in opposite directions with speed 6 (m s^{-1}) collide and rebound with the same speed. What is the impulse imparted to each ball due to the other ?
- Q4.19: A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the muzzle speed of the shell is 80 (m s^{-1}), what is the recoil speed of the gun ?
- Q4.2(a): A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, during its upward motion,
- Q4.2(b): A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, during its downward motion,
- Q4.2(c): A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of 45(^{\circ}) with the horizontal direction? Ignore air resistance.
- Q4.20: A batsman deflects a ball by an angle of 45(^{\circ}) without changing its initial speed which is equal to 54 km/h. What is the impulse imparted to the ball ? (Mass of the ball is 0.15 kg.)
- Q4.21: 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 ?
- Q4.22: If the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks :
- Q4.23(a): Explain why a horse cannot pull a cart and run in empty space,
- Q4.23(b): Explain why passengers are thrown forward from their seats when a speeding bus stops suddenly,
- Q4.23(c): Explain why it is easier to pull a lawn mower than to push it,
- Q4.23(d): Explain why a cricketer moves his hands backwards while holding a catch.
- Q4.3(a): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, just after it is dropped from the window of a stationary train,
- Q4.3(b): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, just after it is dropped from the window of a train running at a constant velocity of 36 km/h,
- Q4.3(c): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, just after it is dropped from the window of a train accelerating with 1 (m s^{-2}),
- Q4.3(d): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, lying on the floor of a train which is accelerating with 1 (m s^{-2}), the stone being at rest relative to the train. Neglect air resistance throughout.
- Q4.4: One end of a string of length (l) is connected to a particle of mass (m) and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed (v) the net force on the particle (directed towards the centre) is : (T) is the tension in the string.
- Q4.5: A constant retarding force of 50 N is applied to a body of mass 20 kg moving initially with a speed of 15 (m s^{-1}). How long does the body take to stop ?
- Q4.6: A constant force acting on a body of mass 3.0 kg changes its speed from 2.0 (m s^{-1}) to 3.5 (m s^{-1}) in 25 s. The direction of the motion of the body remains unchanged. What is the magnitude and direction of the force ?
- Q4.7: A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction of the acceleration of the body.
- Q4.8: The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 4.0 s just in time to save the child. What is the average retarding force on the vehicle ? The mass of the three-wheeler is 400 kg and the mass of the driver is 65 kg.
- Q4.9: A rocket with a lift-off mass 20,000 kg is blasted upwards with an initial acceleration of 5.0 (m s^{-2}). Calculate the initial thrust (force) of the blast.
- Q5.1(a): Give the magnitude and direction of the net force acting on (a) a drop of rain falling down with a constant speed,
- Q5.1(b): Give the magnitude and direction of the net force acting on (b) a cork of mass 10 g floating on water,
- Q5.1(c): Give the magnitude and direction of the net force acting on (c) a kite skillfully held stationary in the sky,
- Q5.1(d): Give the magnitude and direction of the net force acting on (d) a car moving with a constant velocity of 30 km/h on a rough road,
- Q5.1(e): Give the magnitude and direction of the net force acting on (e) a high-speed electron in space far from all material objects, and free of electric and magnetic fields.
- Q5.10: A body of mass 0.40 kg moving initially with a constant speed of $10 m s^{-1}$ to the north is subject to a constant force of 8.0 N directed towards the south for 30 s. Take the instant the force is applied to be $t = 0$, the position of the body at that time to be $x = 0$, and predict its position at $t = –5 s$, $25 s$, $100 s$.
- Q5.11(a): A truck starts from rest and accelerates uniformly at $2.0 m s^{-2}$. At $t = 10 s$, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the (a) velocity of the stone at $t = 11s$ ? (Neglect air resistance.)
- Q5.11(b): A truck starts from rest and accelerates uniformly at $2.0 m s^{-2}$. At $t = 10 s$, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the (b) acceleration of the stone at $t = 11s$ ? (Neglect air resistance.)
- Q5.12(a): A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is $1 m s^{-1}$. What is the trajectory of the bob if the string is cut when the bob is (a) at one of its extreme positions,
- Q5.12(b): A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is $1 m s^{-1}$. What is the trajectory of the bob if the string is cut when the bob is (b) at its mean position.
- Q5.13(a): A man of mass 70 kg stands on a weighing scale in a lift which is moving (a) upwards with a uniform speed of $10 m s^{-1}$,
- Q5.13(b): A man of mass 70 kg stands on a weighing scale in a lift which is moving (b) downwards with a uniform acceleration of $5 m s^{-2}$,
- Q5.13(c): A man of mass 70 kg stands on a weighing scale in a lift which is moving (c) upwards with a uniform acceleration of $5 m s^{-2}$. What would be the readings on the scale in each case?
- Q5.13(d): A man of mass 70 kg stands on a weighing scale in a lift which is moving (d) What would be the reading if the lift mechanism failed and it hurtled down freely under gravity ?
- Q5.14(a): Figure 5.16 shows the position-time graph of a particle of mass 4 kg. What is the (a) force on the particle for $t < 0$, $t > 4 s$, $0 < t < 4 s$?
- Q5.14(b): Figure 5.16 shows the position-time graph of a particle of mass 4 kg. What is the (b) impulse at $t = 0$ and $t = 4 s$ ? (Consider one-dimensional motion only).
- Q5.15(i): Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string. A horizontal force $F = 600 N$ is applied to (i) A, along the direction of string. What is the tension in the string in each case?
- Q5.15(ii): Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string. A horizontal force $F = 600 N$ is applied to (ii) B along the direction of string. What is the tension in the string in each case?
- Q5.16: Two masses 8 kg and 12 kg are connected at the two ends of a light inextensible string that goes over a frictionless pulley. Find the acceleration of the masses, and the tension in the string when the masses are released.
- Q5.17: A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.
- Q5.18: Two billiard balls each of mass 0.05 kg moving in opposite directions with speed $6 m s^{-1}$ collide and rebound with the same speed. What is the impulse imparted to each ball due to the other ?
- Q5.19: A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the muzzle speed of the shell is $80 m s^{-1}$, what is the recoil speed of the gun ?
- Q5.2(a): A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, (a) during its upward motion,
- Q5.2(b): A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, (b) during its downward motion,
- Q5.2(c): A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble, (c) at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of 45° with the horizontal direction? Ignore air resistance.
- Q5.20: A batsman deflects a ball by an angle of 45° without changing its initial speed which is equal to 54 km/h. What is the impulse imparted to the ball ? (Mass of the ball is 0.15 kg.)
- Q5.21: 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 ?
- Q5.23(a): Explain why (a) a horse cannot pull a cart and run in empty space,
- Q5.23(b): Explain why (b) passengers are thrown forward from their seats when a speeding bus stops suddenly,
- Q5.23(c): Explain why (c) it is easier to pull a lawn mower than to push it,
- Q5.23(d): Explain why (d) a cricketer moves his hands backwards while holding a catch.
- Q5.3(a): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, (a) just after it is dropped from the window of a stationary train,
- Q5.3(b): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, (b) just after it is dropped from the window of a train running at a constant velocity of 36 km/h,
- Q5.3(c): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, (c) just after it is dropped from the window of a train accelerating with $1 m s^{-2}$,
- Q5.3(d): Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg, (d) lying on the floor of a train which is accelerating with $1 m s^{-2}$, the stone being at rest relative to the train. Neglect air resistance throughout.
- Q5.4: One end of a string of length l is connected to a particle of mass m and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed v the net force on the particle (directed towards the centre) is : $T$ is the tension in the string. [Choose the correct alternative].
- Q5.5: A constant retarding force of 50 N is applied to a body of mass 20 kg moving initially with a speed of $15 m s^{-1}$. How long does the body take to stop ?
- Q5.6: A constant force acting on a body of mass 3.0 kg changes its speed from $2.0 m s^{-1}$ to $3.5 m s^{-1}$ in 25 s. The direction of the motion of the body remains unchanged. What is the magnitude and direction of the force ?
- Q5.7: A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction of the acceleration of the body.
- Q5.8: The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 4.0 s just in time to save the child. What is the average retarding force on the vehicle ? The mass of the three-wheeler is 400 kg and the mass of the driver is 65 kg.
- Q5.9: A rocket with a lift-off mass 20,000 kg is blasted upwards with an initial acceleration of $5.0 m s^{-2}$. Calculate the initial thrust (force) of the blast.
CBSE Solutions for Class 11 Physics Laws of Motion
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