Q1. A 24g mass rests on a surface and is connected via a string and pulley to a suspended 22g mass. If the suspended mass is allowed to fall, what is the coefficient of friction of the surface? (Assume negligible mass and friction from the string and pulley.)

Background

Topic: Newton's Laws, Friction, and Atwood Machine

This question tests your understanding of forces, friction, and the dynamics of connected masses (an Atwood machine with friction).

Key Terms and Formulas

• Newton's Second Law:

• Frictional Force:

• Normal Force on horizontal surface:

• System acceleration and force analysis for connected masses

Step-by-Step Guidance

1. Draw a free-body diagram for each mass. Label all forces acting on the 24g mass (on the surface) and the 22g mass (hanging).

2. Write Newton's Second Law for each mass. For the 24g mass, consider tension and friction. For the 22g mass, consider tension and gravity.

3. Express the frictional force as , where (convert to kg).

4. Set up the equations for the system, assuming the system just starts to move (acceleration is not zero, but you can relate the forces).

5. Combine the equations to solve for , but stop before plugging in the numbers.

Try solving on your own before revealing the answer!

Q2a. What is the average force required to move a 50kg box up a 20ft ramp (3ft high) into a truck?

Background

Topic: Inclined Planes, Work and Energy, Forces

This question tests your ability to analyze forces on an inclined plane and calculate the force needed to move an object up the ramp at constant velocity.

Key Terms and Formulas

• Inclined Plane Angle:

• Force parallel to ramp:

• Convert units: 1 ft = 0.3048 m

Step-by-Step Guidance

1. Convert the ramp's height and length from feet to meters.

2. Calculate the angle of the ramp using .

3. Write the expression for the force required to push the box up the ramp at constant velocity: .

4. Substitute the values for , , and into the formula, but do not calculate the final value yet.

Try solving on your own before revealing the answer!

Q2b. What angle would double the required force to move the box up the ramp?

Background

Topic: Inclined Planes, Trigonometry, Forces

This question tests your understanding of how the angle of an incline affects the force required to move an object upward.

Key Terms and Formulas

• Force on incline:

• Relationship: To double the force, set

Step-by-Step Guidance

1. Let be the original angle. Express the original force as .

2. Set up the equation for the new angle: .

3. Simplify to .

4. Stop here and encourage the student to solve for .

Try solving on your own before revealing the answer!

Q3a. What is the mass of the acrobat if the rope breaks 9m above the net, causing the net to sink 3.9m? (The net is square, held by 25 identical springs per side, each with N/m.)

Background

Topic: Conservation of Energy, Springs (Hooke's Law)

This question tests your ability to apply energy conservation to a falling mass and a spring system.

Key Terms and Formulas

• Potential Energy:

• Spring Potential Energy:

• Total spring constant:

• Energy conservation:

Step-by-Step Guidance

1. Calculate the total spring constant: N/m (since there are 25 springs per side and 4 sides).

2. Set up the energy conservation equation: , where m and m.

3. Rearrange the equation to solve for .

4. Stop before plugging in the numbers.

Try solving on your own before revealing the answer!

Q3b. How far would the acrobat have fallen before hitting the net if the netting were depressed to 4.9m instead of 3.9m?

Background

Topic: Conservation of Energy, Springs

This question tests your ability to relate the energy stored in the springs to the distance fallen before contact with the net.

Key Terms and Formulas

• Energy conservation:

• Here, m (new depression), as before, as found in part (a).

Step-by-Step Guidance

1. Write the energy conservation equation: .

2. Plug in m and the previously found and .

3. Solve for (the height fallen before hitting the net), but do not calculate the final value yet.

Try solving on your own before revealing the answer!

Q4a. Two cars collide: Your car (4000 lbs, 45 mph) and a stranger's car (3750 lbs, 25 mph) coming from the left. What is the magnitude and direction of the collision?

Background

Topic: Conservation of Momentum, Two-Dimensional Collisions

This question tests your ability to analyze a two-dimensional collision using conservation of momentum.

Key Terms and Formulas

• Momentum:

• Conservation of Momentum:

• Convert units:

• Resultant magnitude:

• Direction:

Step-by-Step Guidance

1. Convert the masses from pounds to kilograms (1 lb = 0.453592 kg) and velocities from mph to m/s.

2. Assign directions: Let your car's motion be along the x-axis and the stranger's car along the y-axis.

3. Calculate the momentum components for each car: and .

4. Find the resultant momentum vector's magnitude and direction using the formulas above, but do not compute the final values yet.

Try solving on your own before revealing the answer!

Q4b. If both cars are moving in the same direction and the stranger rear-ends you at 65 mph, what is your new velocity in mph after the collision?

Background

Topic: Conservation of Momentum, One-Dimensional Collisions

This question tests your ability to apply conservation of momentum to a rear-end collision (one-dimensional).

Key Terms and Formulas

• Conservation of Momentum: (assuming a perfectly inelastic collision)

• Solve for final velocity:

• Convert all units to SI (kg, m/s), then convert the final velocity back to mph.

Step-by-Step Guidance

1. Convert the masses and velocities to SI units.

2. Set up the conservation of momentum equation for a perfectly inelastic collision.

3. Rearrange to solve for (final velocity).

4. Stop before plugging in the numbers and converting back to mph.

Try solving on your own before revealing the answer!