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A cart is driven by a hanging weight of mass m connected via a rope wound around a rear axle (radius r2) that passes over a pulley. The four wheels each have mass m1 and radius r1; the two axles each have mass m2 and radius r2; all are solid cylinders. The cart body mass is negligible. The weight is released from rest at height h above the cart and is guided vertically. Which of the following statements about the subsequent motion are correct?

1. The acceleration of the cart is m*g*r2 / (r1 * [m*(1 + r2²/r1²) + 6*m1 + m2*(2 + r2²/r1²)])
2. The speed of the cart just before the weight lands is sqrt(2*m*g*h / [m*(1 + r2²/r1²) + 6*m1 + m2*(2 + r2²/r1²)])
3. The tension in the rope while the weight falls is m*(g - a*r2/r1), where a is the cart acceleration
4. The tension in the rope while the weight falls is m*(g - a*r2²/r1²), where a is the cart acceleration

Correct answer: The tension in the rope while the weight falls is m*(g - a*r2/r1), where a is the cart acceleration

Solution

The kinematic link between cart speed v and weight speed is v_w = v * r2/r1. Newton's law on the weight gives the tension. Energy methods confirm the acceleration formula.

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