Advertisements
Advertisements
प्रश्न
A 6⋅5 m long ladder rests against a vertical wall reaching a height of 6⋅0 m. A 60 kg man stands half way up the ladder.
- Find the torque of the force exerted by the man on the ladder about the upper end of the ladder.
- Assuming the weight of the ladder to be negligible as compared to the man and assuming the wall to be smooth, find the force exerted by the ground on the ladder.
Advertisements
उत्तर
Given
Mass of the man = m = 60 kg
Ladder length = 6.5 m
Height of the wall = 6 m
(a) We have to find the torque due to the weight of the body about the upper end of the ladder.
\[\tau = 60 \times 10 \times \frac{6 . 5}{2}\sin\theta\]
\[ \Rightarrow \tau = 600 \times \frac{6 . 5}{2} \times \sqrt{\left( 1 - \cos^2 \theta \right)}\]
\[ \Rightarrow \tau = 600 \times \left( \frac{6 . 5}{2} \right) \times \sqrt{\left\{ 1 - \left( \frac{6}{6 . 5} \right)^2 \right\}}\]
\[ \Rightarrow \tau = 740 N - m\]
(b) Let us find the vertical force exerted by the ground on the ladder.
\[N_2 = mg = 60 \times 9 . 8 = 588 N\]
Vertical force exerted by the ground on the ladder = \[\mu N_2 = N_1\]
As system is in rotational equilibrium, we have
\[\tau_{\text{net}} = 0 ............\left(\text{about O} \right)\]
\[\Rightarrow 6 . 5 N_1 \cos\theta = 60g \times \frac{6 . 5}{2}\sin\theta\]
\[\Rightarrow N_1 = \frac{1}{2}60g\tan\theta\]
\[= \frac{1}{2}60g \times \left( \frac{2 . 5}{6} \right) ...........\left[\text{using, }\tan\theta = \frac{2.5}{6} \right]\]
\[ \Rightarrow N_1 = \frac{25}{2}g\]
\[ \Rightarrow N_1 = 122 . 5 N \approx 120 N\]
APPEARS IN
संबंधित प्रश्न
Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.
A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s–1. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of the angular momentum of the cylinder about its axis?
Explain why friction is necessary to make the disc in Figure roll in the direction indicated
(a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(b) What is the force of friction after perfect rolling begins?
The torque of a force \[\overrightarrow F \] about a point is defined as \[\overrightarrow\Gamma = \overrightarrow r \times \overrightarrow F.\] Suppose \[\overrightarrow r, \overrightarrow F\] and \[\overrightarrow \Gamma\] are all nonzero. Is \[r \times \overrightarrow\Gamma || \overrightarrow F\] always true? Is it ever true?
A heavy particle of mass m falls freely near the earth's surface. What is the torque acting on this particle about a point 50 cm east to the line of motion? Does this torque produce any angular acceleration in the particle?
The density of a rod gradually decreases from one end to the other. It is pivoted at an end so that it can move about a vertical axis though the pivot. A horizontal force F is applied on the free end in a direction perpendicular to the rod. The quantities, that do not depend on which end of the rod is pivoted, are ________________ .
A simple pendulum of length l is pulled aside to make an angle θ with the vertical. Find the magnitude of the torque of the weight ω of the bob about the point of suspension. When is the torque zero?
A particle is moving with a constant velocity along a line parallel to the positive X-axis. The magnitude of its angular momentum with respect to the origin is, ______
A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N?
Two discs of the same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of the disc with angular velocities ω1 and ω2. They are brought in to contact face to face coinciding with the axis of rotation. The expression for loss of energy during this process is, ______
A Merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed ω. A person of mass M is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards is ______.
Choose the correct alternatives:
- For a general rotational motion, angular momentum L and angular velocity ω need not be parallel.
- For a rotational motion about a fixed axis, angular momentum L and angular velocity ω are always parallel.
- For a general translational motion , momentum p and velocity v are always parallel.
- For a general translational motion, acceleration a and velocity v are always parallel.
The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?
- The forces may be acting radially from a point on the axis.
- The forces may be acting on the axis of rotation.
- The forces may be acting parallel to the axis of rotation.
- The torque caused by some forces may be equal and opposite to that caused by other forces.
A uniform cube of mass m and side a is placed on a frictionless horizontal surface. A vertical force F is applied to the edge as shown in figure. Match the following (most appropriate choice):
| (a) mg/4 < F < mg/2 | (i) Cube will move up. |
| (b) F > mg/2 | (ii) Cube will not exhibit motion. |
| (c) F > mg | (iii) Cube will begin to rotate and slip at A. |
| (d) F = mg/4 | (iv) Normal reaction effectively at a/3 from A, no motion. |
The position vector of 1 kg object is `vecr = (3hati - hatj)` m and its velocity `vecv = (3hati + hatk)` ms−1. The magnitude of its angular momentum is `sqrtx` Nm where x is ______.
