PH101: Tutorial-4

Center of mass and Momentum

Problem 1: Find the center of mass of the system of objects shown in Fig.1 by

dark boundaries. The objects are cut outs as circular arcs from a thin uniform

plate with specified dimensions.

Fig.1

Problem 2: Consider two interacting charged particles, having the same mass m

and charges q1 and q2, moving in a uniform magnetostatic field ��� (neglect

radiation and gravity). (a) Suppose the charges are identical (q1 = q2 = q), write

the equations of motion in centre-of-mass ��� and relative position �� = ��� − ��

coordinate systems where �� and ��� are the position vectors of the charges q1

and q2. (b) Is it possible to separate the equations of motions in centre-of-mass

��� and relative position �� coordinate systems if the charges have opposite signs

(q1 = −q2 = q)?

Problem 3: A person of mass M is standing on a cart of mass � and the cart is

free to move on its wheels without friction. The person throws a ball of mass m with a speed � at an angle � with the horizontal as measured by the person in

the cart. (a) What are the final velocities of the ball and the cart as seen by an

observer fixed to the ground? (b) With what angle, with respect to the

horizontal, does the fixed observer see the ball leave the cart?

b/2 a/2

y

x

b a O

Problem 5: A raindrop of initial mass M0 starts falling from rest under the

influence of gravity. Assume that the drop gains mass from the cloud at a rate

proportional to the product of its instantaneous mass and its instantaneous

velocity: dM/dt = kMV where k is a constant. Find the speed of the raindrop as a

function of time t and give an expression for the terminal speed. Neglect air

resistance. [Hint: ���

�����=

�tanh�

�

�� �]

Problem 6: An open link chain of length L is coiled up on the edge of a table. A

very small length at one end is pushed off the edge and it starts falling under

gravity. Assume that the velocity of each element of the chain remains zero

until it is jerked into motion with the velocity of the falling section. Find the

velocity when the entire length L has fallen off the table.

Problem 7: A two stage rocket as shown in Fig.3 has the following parameters:

Exhaust speed u, mass of the frame and engine is M1 for the first stage and it is

M2 for the second stage, mass of the fuel is m1 for the first stage and it is m2 for

the second stage. After burning the first stage fuel, M1 drops off. The rocket is

in free space. If the initial velocity is zero, find the final speed achieved.

Compare the speed achieved for a single stage rocket of mass M1+M2+m1+m2

and fuel mass m1+m2.

Fig.3

Problem 4: Material is blown into an

empty cart A from cart B at a rate b. The

material leaves the chute vertically

downward, so that it has the same

horizontal velocity u as cart B. Initially the

cart A of mass M0 was at rest. At the

moment of interest, cart A has mass M and

velocity v, as shown in Fig.2. Find dv/dt,

the instantaneous acceleration of A.

Fig.2

m2 M2 M1 m1