1. The problem asks which statement is true about the center of mass motion of two blocks during a collision. 2. The center of mass velocity of a system with no external forces remains constant due to conservation of momentum. 3. Since there is no friction, the center of mass moves at a constant velocity. 4. Therefore, the correct answer is b: The center of mass moves at a constant velocity of +1.0 m/s because there is no friction acting on the system. 1. The problem asks for the kinetic energy of a box sliding down a ramp with friction. 2. The initial potential energy is $mgh$. 3. Work done against friction is $\mu_k mgL \cos(\theta)$. 4. Kinetic energy at bottom is initial potential energy minus work done by friction: $$KE = mgh - \mu_k mgL \cos(\theta)$$ 5. The correct answer is c. 1. The problem asks which kinetic energy graph corresponds to the gravitational potential energy graph of a falling object. 2. Total mechanical energy is conserved, so kinetic energy increases as potential energy decreases. 3. Kinetic energy starts at zero and increases, curving concave down as velocity increases but acceleration decreases. 4. The correct graph is c. 1. The problem asks for the order of densities of substances given weight and volume. 2. Density $\rho = \frac{\text{mass}}{\text{volume}} = \frac{\text{weight}/g}{\text{volume}}$. 3. Calculate approximate densities: - Gold: $\frac{20/9.8}{0.000106} \approx 19,200$ kg/m$^3$ - Silver: $\frac{50/9.8}{0.000486} \approx 10,500$ kg/m$^3$ - Copper: $\frac{40/9.8}{0.000457} \approx 8,900$ kg/m$^3$ - Iron: $\frac{30/9.8}{0.000389} \approx 7,900$ kg/m$^3$ 4. Order greatest to least: Gold, Silver, Copper, Iron. 5. Correct answer is d. 1. The problem asks for average density of mixture of fluids X and Y. 2. Total mass: $\rho_0 V + \frac{3\rho_0}{2} 2V = \rho_0 V + 3 \rho_0 V = 4 \rho_0 V$ 3. Total volume: $V + 2V = 3V$ 4. Average density: $$\rho_{avg} = \frac{4 \rho_0 V}{3V} = \frac{4}{3} \rho_0$$ 5. Since $\frac{4}{3} \rho_0$ is between $\rho_0$ and $2 \rho_0$, correct answer is c. 1. The problem asks for change in gravitational potential energy of pendulum from max height to position with speed 2.0 m/s. 2. Use conservation of energy: $$\Delta U = \frac{1}{2} m v^2 = \frac{1}{2} \times 0.5 \times 2^2 = 1.0 \text{ J}$$ 3. Correct answer is a. 1. The problem asks for kinetic energy of satellite in circular orbit. 2. Total energy $E = -\frac{GMm}{2R}$, kinetic energy is positive half of magnitude of potential energy: $$KE = \frac{1}{2} \frac{GMm}{R}$$ 3. Correct answer is b. 1. The problem asks for net torque magnitude on wheel slowing down. 2. Angular acceleration: $$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{0 - 3}{6} = -0.5 \text{ rad/s}^2$$ 3. Torque: $$\tau = I \alpha = 2.0 \times 0.5 = 1.0 \text{ N·m}$$ 4. Correct answer is b. 1. The problem asks which pair of forces produces greatest torque about point O. 2. Torque magnitude depends on force magnitude and lever arm. 3. Forces 4 and 5 are applied at ends with largest lever arms and perpendicular directions. 4. Correct answer is d. 1. The problem asks how kinetic energy changes after collision. 2. Inelastic collision with same final velocity means kinetic energy decreases. 3. Correct answer is a. 1. The problem asks for ratio of fluid speeds in pipe sections with radii R and 2R. 2. Continuity equation: $$A_1 v_1 = A_2 v_2$$ $$\pi R^2 v_1 = \pi (2R)^2 v_2 = 4 \pi R^2 v_2$$ $$v_2 = \frac{v_1}{4}$$ 3. Correct answer is a. 1. The problem asks which step is required to ensure valid data points for density measurement. 2. To avoid buoyant force errors, objects must not touch container bottom. 3. Correct answer is c. 1. The problem asks for upward force on rock submerged in oil 3/4 as dense as water. 2. Buoyant force proportional to fluid density. 3. Upward force is $\frac{3}{4} F_0$. 4. Correct answer is a. 1. The problem asks for work done pushing 30 kg object up 7 m incline at 20° with force 150 N. 2. Work: $$W = F d = 150 \times 7 = 1050 \text{ J}$$ 3. Correct answer is b. 1. The problem asks about mechanical energy change of object moving up ramp. 2. Mechanical energy increases because gravitational potential energy increases. 3. Correct answer is a. 1. The problem asks how rate of change of gravitational potential energy compares for two objects of different masses moving at same speed up incline. 2. Rate of change proportional to mass: $$\frac{\Delta U_g}{\Delta t} = mgv \sin(\theta)$$ 3. Since $m_X > m_Y$, rate for X is greater. 4. Correct answer is c. 1. The problem asks for tension in string at lowest point of pendulum with mass 3 kg and length 0.8 m. 2. Speed at lowest point from conservation of energy: $$v = \sqrt{2gL} = \sqrt{2 \times 9.8 \times 0.8} \approx 4 \text{ m/s}$$ 3. Tension: $$T = mg + \frac{mv^2}{L} = 3 \times 9.8 + \frac{3 \times 4^2}{0.8} = 29.4 + 60 = 89.4 \approx 90 \text{ N}$$ 4. Correct answer is d. 1. The problem asks for coefficient of kinetic friction given initial speed 25 m/s, angle 30°, and time 2 s to stop. 2. Acceleration along ramp: $$a = \frac{v}{t} = \frac{25}{2} = 12.5 \text{ m/s}^2$$ 3. Forces along ramp: $$a = g \sin 30° - \mu_k g \cos 30°$$ $$12.5 = 9.8 \times 0.5 - \mu_k \times 9.8 \times 0.866$$ $$12.5 = 4.9 - 8.49 \mu_k$$ $$8.49 \mu_k = 4.9 - 12.5 = -7.6$$ Negative friction impossible, so friction must be zero. 4. Correct answer is a. 1. The problem asks for time block comes to rest with mass 2 kg, speed 12 m/s, and friction coefficient 0.35. 2. Acceleration: $$a = \mu_k g = 0.35 \times 9.8 = 3.43 \text{ m/s}^2$$ 3. Time to stop: $$t = \frac{v}{a} = \frac{12}{3.43} \approx 3.5 \text{ s}$$ 4. Closest answer is b. 1. The problem asks which free body diagram represents forces on block Y hanging from strings. 2. Block Y has tension upward from string above, tension downward from string below, and gravity downward. 3. Correct answer is c. 1. The problem asks for initial speed of ball dropped horizontally from 10 m height landing 50 m away. 2. Time to fall: $$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 10}{9.8}} \approx 1.43 \text{ s}$$ 3. Initial speed: $$v = \frac{50}{1.43} \approx 35 \text{ m/s}$$ 4. Correct answer is c. 1. The problem asks which sphere reaches floor first when rolled off table with same speed. 2. Both fall freely with same vertical acceleration. 3. Both hit floor simultaneously. 4. Correct answer is d.