Section 2.8: Problem-solving Strategies

In solving problems involving forces and motion, adopting a systematic approach improves accuracy and understanding. Key strategies include:

  • Identifying the object(s) of interest.
  • Drawing a clear free-body diagram.
  • Resolving forces into components when necessary.
  • Applying Newton's laws in the appropriate directions.
  • Choosing a suitable coordinate system.
  • Checking units and consistency.
  • Considering limiting cases (e.g., zero friction, maximum tension).
  • Double-checking answers physically (does it make sense?).

Example 1

A 5 kg block is on a frictionless incline of 30° and connected to a hanging mass of 2 kg via a pulley. Find the acceleration of the system and the tension in the string.

Step 1: Draw free-body diagrams for both masses.

Step 2: Write Newton's second law for each mass:

Block on incline: \( T - m_1 g \sin\theta = m_1 a \)

Hanging mass: \( m_2 g - T = m_2 a \)

Step 3: Solve simultaneously:

\( a = \frac{m_2 g - m_1 g \sin\theta}{m_1 + m_2} = \frac{2*9.8 - 5*9.8*0.5}{5+2} = 0.43 \text{ m/s²} \)

\( T = m_1 g \sin\theta + m_1 a = 5*9.8*0.5 + 5*0.43 = 25.65 \text{ N} \)

Example 2

A 10 kg box is pulled along a horizontal surface with coefficient of kinetic friction μ = 0.2 by a 50 N force. Find its acceleration.

Step 1: Identify forces: \( F_{applied} = 50 N \), \( F_{friction} = μ m g = 0.2*10*9.8 = 19.6 N \)

Step 2: Net force: \( F_{net} = F_{applied} - F_{friction} = 50 - 19.6 = 30.4 N \)

Step 3: Acceleration: \( a = F_{net}/m = 30.4 /10 = 3.04 \text{ m/s²} \)

Practice Problems

  1. A 4 kg block slides down a 25° frictionless incline. Find its acceleration.
  2. Two masses, 3 kg and 5 kg, are connected over a pulley. Find acceleration and tension.
  3. A 6 kg box is on a rough horizontal surface with μ=0.3. Pulled with 40 N, find acceleration.
  4. A 2 kg mass is hanging from a string connected to a 3 kg mass on a frictionless incline. Find acceleration.
  5. A 5 kg block is connected to a 2 kg hanging mass with friction μ=0.1. Find tension and acceleration.
  6. Two blocks (3 kg and 7 kg) are on a frictionless surface connected by a rope. Pulling with 20 N, find acceleration and tension.
  7. A 1 kg mass is suspended, and a 2 kg mass is on a horizontal frictionless table. Find system acceleration.
  8. A 4 kg box slides down a 30° incline with friction μ=0.2. Compute acceleration.
  9. A 3 kg mass hangs while 5 kg is on a 45° frictionless incline. Find tension in string.
  10. Two blocks, 6 kg and 4 kg, connected over a pulley with friction μ=0.1. Find acceleration.
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