Physics Problem: Particle Motion in a Horizontal Circle

Given:

• Mass of the particle: 2 kg
• Length of the string: 50 cm
• Radius of the circle: 40 cm
• The particle moves in a horizontal circle with a constant angular speed.
• Center of the circle is vertically below point A.

Objective:

• Calculate the tension in the string.
• Determine the angular speed of the particle.

Concepts to Apply:

• Centripetal Force:

  • The centripetal force required to keep the particle moving in a circle is provided by the horizontal component of the tension in the string.

• Angular Speed (ω):

  • Relationship between linear speed (v) and angular speed given by: ( v = ωr )

• Tension in the String:

  • The tension has both horizontal and vertical components.
  • The vertical component of the tension balances the gravitational force acting on the particle.

Calculations:

1. Vertical Component of Tension (T_vertical):

   • Balances the gravitational force on the particle: [ T_{vertical} = mg ]
   • Where:
     • ( m = 2 \text{ kg} ) (mass of the particle)
     • ( g = 9.8 \text{ m/s}^2 ) (acceleration due to gravity)
     • ( T_{vertical} = 2 \times 9.8 = 19.6 \text{ N} )

2. Horizontal Component of Tension (T_horizontal):

   • Provides the centripetal force for circular motion: [ T_{horizontal} = \frac{mv^2}{r} ]

3. Finding the Total Tension (T):

   • Use Pythagorean theorem: [ T = \sqrt{T_{vertical}^2 + T_{horizontal}^2} ]

4. Calculate Angular Speed (ω):

   • Using the relationship between linear speed, radius, and angular speed: [ v = ωr ]
   • Rearrange to find ( ω ): [ ω = \frac{v}{r} ]_

Conclusion:

• By assessing the balance of forces and the relationship between linear and angular speed, the tension in the string and the angular speed of the particle can be calculated using the principles of mechanics and circular motion.