Differential Equations in Physics
This quiz covers the applications of differential equations in physics, including topics such as motion, heat transfer, and oscillations.
Questions
A particle moves in a straight line with a velocity given by the function (v(t) = 3t^2 - 2t + 1). What is the particle's acceleration at time (t = 2) seconds?
- \(4\)
- \(6\)
- \(8\)
- \(10\)
A mass of 10 kg is attached to a spring with a spring constant of 100 N/m. The mass is pulled 5 cm to the right of its equilibrium position and released. What is the equation of motion for the mass?
- \(mx'' + kx = 0\)
- \(mx'' - kx = 0\)
- \(mx'' + kx = 10\)
- \(mx'' - kx = 10\)
A metal rod of length (L) is heated at one end so that the temperature at a distance (x) from the heated end is given by the function (T(x) = 100 - 20x). What is the rate of heat flow through the rod at a distance (x = 2) meters from the heated end?
- \(-40\) W
- \(-20\) W
- \(20\) W
- \(40\) W
A pendulum of length (L) is released from an angle (\theta_0). What is the equation of motion for the pendulum?
- \(m\frac{d^2\theta}{dt^2} + mg\sin\theta = 0\)
- \(m\frac{d^2\theta}{dt^2} - mg\sin\theta = 0\)
- \(m\frac{d^2\theta}{dt^2} + mg\cos\theta = 0\)
- \(m\frac{d^2\theta}{dt^2} - mg\cos\theta = 0\)
A circuit consists of a resistor of resistance (R), an inductor of inductance (L), and a capacitor of capacitance (C). The charge on the capacitor is given by the function (q(t) = Q_0\cos\omega t), where (Q_0) is the initial charge on the capacitor and (\omega) is the angular frequency of the circuit. What is the equation of motion for the charge on the capacitor?
- \(LC\frac{d^2q}{dt^2} + RC\frac{dq}{dt} + \frac{1}{C}q = 0\)
- \(LC\frac{d^2q}{dt^2} - RC\frac{dq}{dt} + \frac{1}{C}q = 0\)
- \(LC\frac{d^2q}{dt^2} + RC\frac{dq}{dt} - \frac{1}{C}q = 0\)
- \(LC\frac{d^2q}{dt^2} - RC\frac{dq}{dt} - \frac{1}{C}q = 0\)