Solving for the coefficients by using the factor theorem
Question: If the polynomial, \(P(x) = ax^4 + (b-a)x^3 + (1-2ab)x^2 + (ab-10)x + 2ab\), has a factor \(x-2\), then […]
When one rotating disk contacts with another disk: Conservation of angular momentum
Question: A solid cylinder with a moment of inertia, \(I_0\), rotates about its center at an angular velocity of \(\omega_0\). […]
If you are standing up on a suspended plane: Torques and the balance
Question: A horizontal beam is attached to a wall. The length and weight are 10.0 m and 200 N, respectively. […]
Quadratic equation 2: Solving it with the discriminant
Question: The equation, \(kx^2 -(k+3)x – 1 =0\), which has real coefficients having complex roots, \(a+ib\) and \(a-ib\). In order […]
A problem for the quadratic equation, using the relationship between solutions and coefficients 1
Question: The roots of \(x^2 – 5x + 3 = 0\) are \(\alpha\) and \(\beta\). Another quadratic equation \(x^2 + […]
The bounded solution of a first order ordinary differential equation
Question: Find the bounded solution of the following ordinary differential equation: \[ \frac{dx}{dt} = 2x + \sin t \] for \(-\infty […]
Find the angle of a horizontal projectile motion!!
Question: A ball is thrown horizontally from the height of 40.0 m. The horizontal range of the projectile is known […]
The derivative of parametric equations
Question: \(x\) and \(y\) are given as follows: \[ x = e^t \cos t \] \[ y = e^t \sin […]
Quantum mechanics: Converting momentum into configuration spaces
Question: In quantum theory, the wave function can be convertible either in momentum space or in configuration space. Suppose we […]
What should we do when complex variables involve trigonometric functions?
Question: Define \(z=x+iy\). Prove the following equation \[ \frac{\sin 2x + i\sinh 2y}{\cos 2x + \cosh 2y} = \tan z \] Answer: […]