Q7. A 1000-N piano is raised at constant speed using a very light rope in a frictionless pulley system (see image). With what force is the mover pulling down on the rope?

Background

Topic: Newton's Laws, Tension, and Pulley Systems

This question tests your understanding of how forces are distributed in a simple frictionless pulley system, especially when lifting an object at constant speed.

Key Terms and Formulas

• Tension (T): The force transmitted through a rope or cable when it is pulled tight by forces acting from opposite ends.

• Newton's Second Law:

• Constant Speed: Implies , so .

• Weight (W):

Step-by-Step Guidance

1. Recognize that the piano is being lifted at constant speed, so the net force on the piano is zero ().

2. Identify the forces acting on the piano: its weight (downward) and the tension in the rope (upward).

3. Set up the force balance equation for the piano: (since ).

4. Consider the pulley system: In a simple frictionless pulley, the force the mover applies downward on the rope equals the tension in the rope.

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Q8. Three blocks (A, B, C) are connected by light ropes and a frictionless pulley as shown. An external force P is applied downward on block A, causing it to accelerate downward at 2.5 m/s². The tension in the rope connecting block B and block C is 60 N. (a) What is the magnitude of the force P? (b) What is the mass of block C?

Background

Topic: Newton's Second Law, Tension, and Pulley Systems

This problem involves analyzing forces and tensions in a multi-block system with a pulley, using Newton's laws to relate forces, masses, and acceleration.

Key Terms and Formulas

• Newton's Second Law:

• Tension (T): The force transmitted through a rope.

• Weight (W):

Step-by-Step Guidance

1. Draw free-body diagrams for each block, labeling all forces (weights, tensions, and applied force P).

2. Write Newton's second law equations for each block, considering the direction of acceleration and the forces acting on each.

3. For block B, use the given tension (60 N) and its mass (18 kg) to set up the equation: (for block C), and (for block B).

4. For block A, relate the applied force P, the weight of A, and the tension in the rope connecting A and B, using .

5. Combine the equations to express P and in terms of known quantities and solve up to the point where you can substitute values.

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Q9. A 10-kg block on a perfectly smooth horizontal table is connected by a horizontal string to a 63-kg block hanging over the edge of the table. What is the magnitude of the acceleration of the 10-kg block when the other block is gently released?

Background

Topic: Newton's Second Law, Atwood Machine (Pulley Systems)

This question tests your ability to analyze a two-mass system connected by a string over a pulley, where one mass hangs and the other is on a frictionless surface.

Key Terms and Formulas

• Newton's Second Law:

• Atwood Machine Acceleration: (for a frictionless system)

Step-by-Step Guidance

1. Identify the forces acting on each block: gravity on the hanging block, tension in the string, and gravity/normal force on the block on the table.

2. Write Newton's second law for each block, considering the direction of acceleration.

3. Set up the system of equations for the two blocks, using the fact that the tension is the same throughout the string and the acceleration is the same for both blocks.

4. Combine the equations to solve for the acceleration in terms of the masses and .

5. Substitute the given masses and m/s² into the formula, but stop before calculating the final value.

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Q10. Two packages are connected by a very light string over an ideal pulley. Package A (3.0 kg) can slide along a rough plane inclined at 30° above the horizontal. The string acts on package A parallel to the surface. The coefficient of static friction between package A and the plane is 0.40. What minimum mass should package B have in order to start package A sliding up the ramp?

Background

Topic: Static Friction, Inclined Planes, Pulley Systems

This question tests your understanding of the forces acting on an object on an inclined plane, including friction, gravity, and tension, and how to determine the minimum force (or mass) needed to overcome static friction.

Key Terms and Formulas

• Static Friction Force:

• Normal Force on Incline:

• Component of Gravity Down the Ramp:

• Force Balance for Motion Up the Ramp: (at threshold of motion)

Step-by-Step Guidance

1. Draw a free-body diagram for package A, showing all forces: gravity, normal force, friction, and tension from the string.

2. Calculate the normal force on package A using .

3. Find the maximum static friction force using .

4. Set up the force balance equation along the ramp, including the tension from package B and the forces resisting motion up the ramp.

5. Express the tension in terms of the weight of package B, and solve for the minimum mass needed, but stop before the final calculation.

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