Q1. Two children push on opposite sides of a door. One pushes with 175 N at 0.700 m from the hinges, the other at 0.400 m. What force must the second exert to keep the door from moving? (Neglect friction.)

Background

Topic: Rotational Equilibrium and Torque

This question tests your understanding of how forces applied at different distances from a pivot (hinge) create torques, and how to balance them for equilibrium.

Key Terms and Formulas:

• Torque ():

• Rotational Equilibrium: (sum of torques must be zero for no rotation)

• , ,

Step-by-Step Guidance

1. Write the condition for rotational equilibrium: (since the door doesn't move).

2. Express the torques: , .

3. Set up the equation: .

4. Rearrange to solve for : .

Try solving on your own before revealing the answer!

Q2. An object of mass kg is attached to a spring ( N/m) and released from rest when the spring is stretched 4 m.

Background

Topic: Hooke's Law and Spring Forces

This question tests your understanding of the force exerted by a spring when stretched or compressed, and how it changes with displacement.

Key Terms and Formulas:

• Hooke's Law:

• = spring constant (N/m)

• = displacement from equilibrium (m)

Step-by-Step Guidance

1. Identify the spring constant N/m and initial stretch m.

2. Apply Hooke's Law: .

3. Calculate the force: .

4. Interpret the sign: The negative sign means the force is directed toward the equilibrium position (opposite the stretch).

Try solving on your own before revealing the answer!

Q2b. What force is exerted when the spring is stretched only 2 m?

Background

Topic: Hooke's Law (continued)

This part asks you to apply the same principle for a different displacement.

Key Terms and Formulas:

• m

• Use

Step-by-Step Guidance

1. Plug in the new displacement: .

2. Again, the negative sign indicates direction toward equilibrium.

Try solving on your own before revealing the answer!

Q2c. At what point does the spring force vanish?

Background

Topic: Equilibrium Position of a Spring

This part tests your understanding of when a spring exerts no force.

Key Terms and Formulas:

• Equilibrium position:

• At ,

Step-by-Step Guidance

1. Recall that .

2. Set to find when .

Try solving on your own before revealing the answer!

Q3. A uniform beam is supported by a cable at one end and friction at the other. The cable makes a 30° angle with the horizontal, the beam is 2.00 m long, the coefficient of static friction is , and the beam's weight is . What is the minimum distance from point A at which an additional weight can be hung without causing the rod to slip?

Background

Topic: Static Equilibrium, Torque, and Friction

This question tests your ability to analyze forces and torques on a beam, including the effects of friction and additional weights, to determine the conditions for equilibrium.

Key Terms and Formulas:

• Torque ():

• Static Friction:

• Equilibrium Conditions: , ,

• Beam length m, , weight , extra weight at distance from A

Step-by-Step Guidance

1. Draw a free-body diagram showing all forces: weight of beam ( at center), extra weight ( at distance ), tension in cable, normal force, and friction at point A.

2. Write the torque equilibrium equation about point A (sum of torques must be zero):

3. Write the force equilibrium equations in the horizontal and vertical directions.

4. Express the maximum static friction force: (where is the normal force at A).

5. Set up the condition for the beam to be on the verge of slipping: the horizontal component of the cable tension must equal .

6. Combine these equations to solve for the minimum where the extra weight can be hung without slipping.

Try solving on your own before revealing the answer!