Advertisements
Advertisements
Question
Consider the situation of the previous problem. Suppose the block of mass m1 is pulled by a constant force F1 and the other block is pulled by a constant force F2. Find the maximum elongation that the spring will suffer.
Solution
Given:
Force on block of mass, m1 = F1
Force on block of mass, m2 = F2
Let the acceleration produced in mass m1 be a1.
\[a_1 = \frac{F_1 - F_2}{m_1 + m_2}\]
Let the acceleration of mass m2 be a2.
\[a_2 = \frac{F_2 - F_1}{m_1 + m_2}\]
Due to the force F2, the mass m1 experiences a pseudo force.
\[\therefore \text{ Net force on m}_1 = F_1 + m_1 a_2 \]
\[F' = F_1 + m_1 \times \frac{( F_2 - F_1 )}{m_1 + m_2}\]
\[ = \frac{m_1 F_1 + m_2 F_1 + m_1 F_2 - m_1 F_1}{m_1 + m_2}\]
\[ = \frac{m_2 F_1 + m_1 F_2}{m_1 + m_2}\]
Similarly, mass m2 experiences a pseudo force due to force F1.
\[\therefore \text{ Net force on m}_2 = F_2 + m_2 a_1\]
\[F " = F_2 + m_2 \times \frac{( F_1 - F_2 )}{m_1 + m_2}\]
\[ = \frac{m_1 F_2 + m_2 F_2 + m_2 F_1 - m_2 F_2}{m_1 + m_2}\]
\[ = \frac{m_1 F_2 + m_2 F_1}{m_1 + m_2}\]
Let m1 be displaced by a distance x1 and m2 be displaced by a distance x2.
Therefore, the maximum elongation of the spring = x1 + x2
Work done by the blocks = Energy stored in the spring
\[\Rightarrow \frac{m_2 F_1 + m_1 F_2}{m_1 + m_2} \times x_1 \times \frac{m_2 F_1 + m_1 F_2}{m_1 + m_2} \times x_2 = \left( \frac{1}{2} \right)k( x_1 + x_2 )^2 \]
\[ \Rightarrow x_1 + x_2 = \frac{2}{k}\left( \frac{m_1 F_2 + m_2 F_1}{m_1 + m_2} \right)\]
APPEARS IN
RELATED QUESTIONS
The centre of mass is defined as \[\vec{R} = \frac{1}{M} \sum_i m_i \vec{r_i}\]. Suppose we define "centre of charge" as \[\vec{R}_c = \frac{1}{Q} \sum_i q_i \vec{r_i}\] where qi represents the ith charge placed at \[\vec{r}_i\] and Q is the total charge of the system.
(a) Can the centre of charge of a two-charge system be outside the line segment joining the charges?
(b) If all the charges of a system are in X-Y plane, is it necessary that the centre of charge be in X-Y plane?
(c) If all the charges of a system lie in a cube, is it necessary that the centre of charge be in the cube?
You are waiting for a train on a railway platform. Your three-year-old niece is standing on your iron trunk containing the luggage. Why does the trunk not recoil as she jumps off on the platform?
If the external force acting on a system have zero resultant, the centre of mass
(a) must not move
(b) must not accelerate
(c) may move
(d) may accelerate.
Two balls are thrown simultaneously in air. The acceleration of the centre of mass of the two balls while in air
Three particles of masses 1.0 kg, 2.0 kg and 3.0 kg are placed at the corners A, B and C respectively of an equilateral triangle ABC of edge 1 m. Locate the centre of mass of the system.
Two blocks of masses 10 kg and 30 kg are placed along a vertical line. The first block is raised through a height of 7 cm. By what distance should the second mass be moved to raise the centre of mass by 1 cm?
A car of mass M is at rest on a frictionless horizontal surface and a pendulum bob of mass m hangs from the roof of the cart. The string breaks, the bob falls on the floor, makes serval collisions on the floor and finally lands up in a small slot made in the floor. The horizontal distance between the string and the slot is L. Find the displacement of the cart during this process.
A ball of mass m is dropped onto a floor from a certain height. The collision is perfectly elastic and the ball rebounds to the same height and again falls. Find the average force exerted by the ball on the floor during a long time interval.
A block of mass 2.0 kg moving 2.0 m/s collides head on with another block of equal mass kept at rest. (a) Find the maximum possible loss in kinetic energy due to the collision. (b) If he actual loss in kinetic energy is half of this maximum, find the coefficient of restitution.
A projectile is fired with a speed u at an angle θ above a horizontal field. The coefficient of restitution of collision between the projectile and the field is e. How far from the starting point, does the projectile makes its second collision with the field?
Consider the situation of the previous problem. Suppose each of the blocks is pulled by a constant force F instead of any impulse. Find the maximum elongation that the spring will suffer and the distance moved by the two blocks in the process.
Solve the following problem.
Four uniform solid cubes of edges 10 cm, 20 cm, 30 cm and 40 cm are kept on the ground, touching each other in order. Locate centre of mass of their system.
In system of two particles of masses 'm1' and 'm2', the first particle is moved by a distance 'd' towards the centre of mass. To keep the centre of mass unchanged, the second particle will have to be moved by a distance ______.
The centre of mass of a system of two particles divides the distance between them ______.
Which of the following has maximum momentum?
A shell of mass 'M' initially at rest suddenly explodes in three fragments. Two of these fragments are of mass 'M/4' each, which move with velocities 3 ms-1 and 4 ms-1 respectively in mutually perpendicular directions. The magnitude of velocity of the third fragment is ______.
Which of the following points is the likely position of the centre of mass of the system shown in figure?
A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind figure. The CM of the plate is now in the following quadrant of x-y plane ______.
The spheres of masses 2 kg and 4 kg are situated at the opposite ends of wooden bars of length 9 m. Where does the centre of mass of the system will ______.
