Advertisements
Advertisements
Question
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.
Solution
The mass of the bob is m.
The mass of the cart is M.
Considering the bob falls at point A.
Initial distance of centre of mass of the system from P is given as
\[x = \frac{m \times L + M \times 0}{M + m} = \frac{m}{M + m}L\]
When the bob falls in the slot, the distance of centre of mass of the system from P becomes zero.
\[\therefore \text{ Shift in centre of mass } = 0 - \frac{mL}{M + m}\]
\[ = - \frac{mL}{M + m} \text{ towards left}\]
\[ = \frac{mL}{M + m} \text{ towards right}\]
Therefore, the cart moves a distance of
\[\frac{mL}{M + m}\] towards right.
APPEARS IN
RELATED QUESTIONS
If all the particles of a system lie in X-Y plane, is it necessary that the centre of mass be in X-Y plane?
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?
In a head-on collision between two particles, is it necessary that the particles will acquire a common velocity at least for one instant?
Consider the following two statements:
(A) Linear momentum of the system remains constant.
(B) Centre of mass of the system remains at rest.
In which of the following cases the centre of mass of a rod is certainly not at its centre?
(a) the density continuously increases from left to right
(b) the density continuously decreases from left to right
(c) the density decreases from left to right upto the centre and then increases
(d) the density increases from left to right upto the centre and then decreases.
A square plate of edge d and a circular disc of diameter d are placed touching each other at the midpoint of an edge of the plate as shown in figure. Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.
Consider a gravity-free hall in which a tray of mass M, carrying a cubical block of ice of mass m and edge L, is at rest in the middle. If the ice melts, by what distance does the centre of mass of "the tray plus the ice" system descend?
A railroad car of mass M is at rest on frictionless rails when a man of mass m starts moving on the car towards the engine. If the car recoils with a speed v backward on the rails, with what velocity is the man approaching the engine?
A particle of mass 100 g moving at an initial speed u collides with another particle of same mass kept initially at rest. If the total kinetic energy becomes 0.2 J after the collision, what could be the minimum and the maximum value of u.
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?
Solve the following problem.
A uniform solid sphere of radius R has a hole of radius R/2 drilled inside it. One end of the hole is at the center of the sphere while the other is at the boundary. Locate center of mass of the remaining sphere.
The centre of mass of a system of two particles divides the distance between them ______.
Which of the following has maximum momentum?
A bullet of mass 20 gram is fired from a gun of mass 2.5 kg with a speed of 750 m/s. The magnitude of recoil velocity of the gun is ______.
In rotational motion of a rigid body, all particles move with ______.
The ratio of weights of a man inside a lift when it is stationary and when it is going down with a uniform acceleration 'a' is 3 : 2. The value of 'a' will be ______.
(a< g, g = acceleration due to gravity)
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
- Show pi = p’i + miV Where pi is the momentum of the ith particle (of mass mi) and p′ i = mi v′ i. Note v′ i is the velocity of the ith particle relative to the centre of mass. Also, prove using the definition of the centre of mass `sum"p""'"_"t" = 0`
-
Show K = K′ + 1/2MV2
where K is the total kinetic energy of the system of particles, K′ is the total kinetic energy of the
system when the particle velocities are taken with respect to the centre of mass and MV2/2 is the
kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the
system). The result has been used in Sec. 7.14. - Show where `"L""'" = sum"r""'"_"t" xx "p""'"_"t"` is the angular momentum of the system about the centre of mass with
velocities taken relative to the centre of mass. Remember `"r"_"t" = "r"_"t" - "R"`; rest of the notation is the standard notation used in the chapter. Note L′ and MR × V can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles. - Show `"dL"^"'"/"dt" = ∑"r"_"i"^"'" xx "dP"^"'"/"dt"`
Further show that `"dL"^'/"dt" = τ_"ext"^"'"`
Where `"τ"_"ext"^"'"` is the sum of all external torques acting on the system about the centre of mass.
(Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
Which of the following points is the likely position of the centre of mass of the system shown in figure?
Figure shows a lamina in x-y plane. Two axes z and z ′ pass perpendicular to its plane. A force F acts in the plane of lamina at point P as shown. Which of the following are true? (The point P is closer to z′-axis than the z-axis.)
- Torque τ caused by F about z axis is along `-hatk`.
- Torque τ′ caused by F about z′ axis is along `-hatk`.
- Torque τ caused by F about z axis is greater in magnitude than that about z axis.
- Total torque is given be τ = τ + τ′.
(n – 1) equal point masses each of mass m are placed at the vertices of a regular n-polygon. The vacant vertex has a position vector a with respect to the centre of the polygon. Find the position vector of centre of mass.
