Figure 8-31 shows an arrangement with an air track, in which...

Figure $8-31$ shows an arrangement with an air track, in which a cart is connected by a cord to a hanging block. The cart has mass $m_{A}=0.600 \mathrm{~kg}$ and its center is initially at $x y$ coordinates $(-0.500 \mathrm{~m}, 0.000 \mathrm{~m}) ;$ the block has mass $m_{B}=0.400$ $\mathrm{kg}$ 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 center of mass as a function of time $t ?(\mathrm{c})$ Sketch the path taken by the system's center of mass. (d) If the path is curved, does it bulge upward to the right or downward to the left? If, instead, it is straight, give the angle between it and the $x$ axis.

Key Concepts

Vector Representation and Unit-Vector Notation

Using vectors to describe physical quantities like acceleration and velocity is essential because these quantities have both magnitude and direction. Unit-vector notation provides a convenient method to express these vectors in terms of their components along coordinate axes, making it easier to analyze and solve problems in two or more dimensions.

Kinematics under Constant Acceleration

Kinematics involves the study of motion without considering its causes. In the context of constant acceleration, kinematic equations can be used to relate displacement, initial velocity, time, and acceleration. This allows one to determine the velocity of the center of mass as a function of time and to predict its trajectory.

Center of Mass

The center of mass is the weighted average position of all the mass in a system. It represents the point at which the entire mass of the system can be considered to be concentrated for the analysis of translational motion. The motion of the center of mass is influenced solely by external forces, regardless of internal forces which cancel by Newton's third law.

Net External Force and Newton's Second Law for a System

Newton's second law, when applied to a system, states that the acceleration of the center of mass is given by the net external force acting on the system divided by the total mass. This concept is crucial for determining how the system as a whole will accelerate, regardless of internal interactions between its parts.

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