Math Solutions

A spring with k = 200 N/m is compressed 0.3 m. Calculate the elastic potential energy stored.

A pendulum swings back and forth. Show how kinetic and potential energy trade off at each point.

A car goes around a banked curve of radius r at speed v with no friction. Find the required banking angle.

A 2 kg ball swings in a horizontal circle of radius 1 m at 5 m/s. Find the centripetal force.

A 20 kg box sits on a flat floor with friction coefficient μ = 0.3. What horizontal force is needed to start it moving?

A 100 N weight hangs from two ropes attached to the ceiling at 30° and 45° from vertical. Find the tensions in both ropes.

A 50 kg box sits on a frictionless 30° ramp. Show the forces and find the acceleration down the slope.

Two forces of 10 N (along +x) and 15 N (at 60° above +x) act on an object. Find the third force needed for equilibrium.

Visualize the cross product A × B and use the right-hand rule to determine its direction in 3D.

Visualize how A · B = |A| |B| cos(θ) represents the projection of one vector onto another.

Decompose a 50 N force at 30° above horizontal into its horizontal and vertical components.

Add vectors A = (3, 4) and B = (2, -1) graphically using the tip-to-tail method.

Graph y = e^(−0.2t) · sin(4t) to show a damped oscillation.

Combine sin(10t) and sin(10.5t) to create a beat pattern. What is the beat frequency?

Show two sine waves 180° out of phase canceling each other to produce destructive interference.

Show two sine waves with same frequency and amplitude combining in phase to produce constructive interference.

Find angle C in a triangle with sides a = 7, b = 9, c = 12.

In triangle ABC, A = 40°, B = 60°, and a = 10. Find b and c.

Show how y = sin(x − π/4) is shifted compared to y = sin(x).

Compare y = sin(x), y = 2 sin(x), and y = sin(2x). How do amplitude and frequency change?