A block of mass $$M_1$$ is connected by a string to a sphere of mass $$M_2$$. The block is on a frictionless plane inclined at an angle of $$30^\circ$$, as shown in the figure above. The string passes over a pulley of negligible mass and friction, and the sphere hangs freely. Which of the following is always true if the block is accelerating down the incline?

Two carts of equal mass are on a horizontal surface with negligible friction. Cart A is approaching cart B, which is at rest, as shown in Figure 1 above. Attached to cart B is a spring that is initially compressed. At the moment cart A collides with cart B, the spring is released and pushes on cart A, as shown in Figure 2 above. Which of the following correctly states what happens to the kinetic energy and the momentum of the two-cart system as a result of the collision compared to those quantities before the collision?

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Particle $$A$$ and particle $$B$$, each of mass $$M$$, move along the x-axis exerting a force on each other. The potential energy of the system of two particles associated with the force is given by the equation $$U=\beta / r^{2}$$ , where $$r$$ is the distance between the two particles and $$\beta$$ is a positive constant. At time $$t = 0$$, particle $$A$$ is located at $$x =2D$$ with an initial speed of v₀ to the left, and particle $$B$$ is at rest at the origin, as shown in the figure above.

At time $$t = T_1$$, particle $$A$$ is observed to be traveling with speed 2v₀ / 3 to the left. The speed and direction of motion of particle B is

A sphere of mass $$m$$ hangs from two strings that each make an angle of $$30^\circ$$ with the horizontal, as shown in the figure above. What is the tension $$T$$ in one of the strings?

A ball of mass 2.0 kg is thrown vertically upward from the top of a building. The ball’s velocity $$v$$ is given as a function of time $$t$$ by the equation $$v(t) = R - St$$ , where R = 5 m/s and S = 10 m/s² . The positive direction is upward.

What is the ball's acceleration at time $$t = 1 \ \text{s}$$?

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A solid disk of mass $$M$$ and radius $$R$$ is freely rotating horizontally in a counterclockwise direction with angular speed ω about a vertical axis through its center with negligible friction. The rotational inertia of the disk is $$MR^{2}/2$$. A second identical disk is at rest and suspended above the first disk with the centers of the two disks aligned, as shown in the figure above. There is no contact between the disks. The second disk is dropped onto the first disk, and after a short time they rotate counterclockwise with the same angular speed $$w_{f}$$.

Which of the following properties of the two-disk system must be conserved between the time the second disk is dropped on the first disk and the time that the two disks begin rotating with the same speed?

A student throws a metal sphere, which follows the path shown above. If air resistance is negligible, which of the following is a correct relationship between the magnitudes of the accelerations $$a_K$$, $$a_L$$, and $$a_M$$ at the three points shown?