Figure 9-44 shows an arrangement with an air track, in which...

Figure 9-44 shows an arrangement with an air track, in which a cart is connected by a cord to a hanging block. The cart has mass $m_{1}=0.600 \mathrm{~kg}$, and its center is initially at $x y$ coordinates $(-0.500$ $\mathrm{m}, 0 \mathrm{~m}) ;$ the block has mass $m_{2}=0.400 \mathrm{~kg}$, and its center is initially at $x y$ coordinates $(0,-0.100 \mathrm{~m})$. The mass of the cord and pulley are negligible. The cart is released from rest, and both cart and block move until the cart hits the pulley. The friction between the cart and the air track and between the pulley and its axle is negligible. (a) In unit-vector notation, what is the acceleration of the center of mass of the cart-block system? (b) What is the velocity of the com as a function of time $t ?$ (c) Sketch the path taken by the com. (d) If the path is curved, determine whether it bulges upward to the right or downward to the left, and if it is straight, find the angle between it and the $x$ axis.

Key Concepts

Center of Mass

The center of mass of a system is the weighted average position of all the mass in the system. It acts as if all the mass were concentrated at that point when analyzing the external forces and motion. Calculating the acceleration or velocity of the center of mass involves summing all external forces over the total mass, independent of the internal forces between the components.

Newton's Second Law for a System

Newton's second law states that the net external force acting on a system equals the total mass of the system times the acceleration of its center of mass. This principle applies to multi-object systems as well, meaning that when internal forces cancel, only external forces determine the motion of the center of mass.

System Analysis and Internal Forces Cancellation

When analyzing a system made up of multiple interconnected parts, it is important to identify which forces are internal and which are external. Internal forces (such as tension in a cord connecting two objects) do not affect the acceleration of the center of mass because they cancel out according to Newton’s third law. This simplifies the analysis by allowing one to consider only the net external forces.

Vector Representation and Unit Vectors

Expressing physical quantities like displacement, velocity, or acceleration in unit-vector notation allows for a clear representation of their direction and magnitude in a chosen coordinate system. This method is crucial for problems involving motion in multiple dimensions, where the decomposition of vectors into components simplifies the analysis.

Kinematics Under Constant Acceleration

Kinematics deals with the motion of objects and is particularly streamlined under conditions of constant acceleration. In such scenarios, the equations of motion provide direct relations between displacement, velocity, time, and acceleration, enabling the derivation of velocity as a function of time and the prediction of the trajectory of the center of mass.

Transcript

00:01 Is this problem? we have a scenario in which a card a card is connected to accord.

00:08 This court is connected to a mass that is hanging here.

00:16 We call this the mosque, which is lying on the cart.

00:19 This mosque is which is a card, actually.

00:23 So this is on air track here.

00:25 So what? so this is in the air trick.

00:27 This is a mass one, and this is mass to also we assume this is forced to x axis, and this is why access in, but area has to find the pictorial.

00:41 So we had us to find, uh, there.

00:45 Victoria ll acceleration the victoria exploration relation for the center mass off the system.

00:55 We can write their pictorial form of exploration for central mass of the system.

01:00 You simply mass one times exploration one plus in two.

01:05 Exhibition too.

01:07 You gotta die some off the masses, one plus into, uh so the net force on the system will be the total moss, uh, times explosion due to gravity, which is a coulter in two org.

01:28 That's from the newton's law.

01:29 We converted exploration will be g, um, into moss two divided by last one, plus moss too well, this gives us the 0.4 times off g for the block one, the exploration is to the right, which is in my direction, direction and for the block, too.

01:57 The explosion is in minus g.

02:01 Our direction then from the equation that we're all dead here we can just absolute values...

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