Concepts Of Physics MCQ Edition [Volume 1]PhysicsWave Motion and Waves on a String
A wave is described by the equation y = (1.0 mm ) ( x 2.0 cm - t 0.01 s ) . What are the speeds of the particles at x = 1.0 cm at t = 0.011 , 0.012 , and 0.013 s ?
Options
- A18 cm s ⁻¹ , 25 cm s ⁻¹ , 31 cm s ⁻¹
- B9.7 cm s ⁻¹ , 18 cm s ⁻¹ , 25 cm s ⁻¹
- C9.7 mm s ⁻¹ , 18 mm s ⁻¹ , 25 mm s ⁻¹
- D5.0 cm s ⁻¹ , 10 cm s ⁻¹ , 15 cm s ⁻¹
Correct answer
B. 9.7 cm s ⁻¹ , 18 cm s ⁻¹ , 25 cm s ⁻¹
Step-by-step solution
The given wave equation is y = (1.0 mm ) ( x 2.0 - 100 t ) . The particle velocity v is given by the partial derivative of y with respect to time t : v = y t = (1.0 mm ) ( x 2.0 - 100 t ) (-100 s ⁻¹) v = -100 ( x 2.0 - 100 t ) mm s ⁻¹ At x = 1.0 cm , the argument of the cosine becomes 2 - 100 t . Using ( 2 - ) = , the velocity simplifies to: v = -100 (100 t) mm s ⁻¹ = -10 (100 t) cm s ⁻¹ The speed is the magnitude of the velocity, |v| = 10 | (100 t)| cm s ⁻¹ . At t = 0.011 s : |v| = 10 | (1.1 )| = 10 (18^ ) = 10 (