Math Solutions

Compare oscillations x(t) = A sin(ωt) for different values of A and ω.

Show how kinetic and potential energy oscillate during SHM, and that their sum stays constant.

Show how a pendulum's period depends on its length, but not its mass or amplitude (for small angles).

A 2 kg mass attached to a spring with k = 50 N/m oscillates. Show position, velocity, and acceleration over time.

A spinning ice skater pulls her arms in. How does her angular velocity change?

Apply a force at varying distances and angles from a pivot. Show how torque changes with lever arm.

Compare the rolling acceleration of a disk, a hoop, and a solid sphere with the same mass and radius down a frictionless incline.

A satellite orbits Earth in a circular orbit at 8000 m/s. Find its orbital radius and period.

A ball on a string swings in a vertical circle of radius 1 m. What minimum speed does it need at the top so that the string stays taut?

A car speeds up at 2 m/s² while turning around a curve of radius 30 m at 10 m/s. Find the total acceleration.

A wheel completes 5 full revolutions in 2 seconds. Find its angular velocity in rad/s, frequency in Hz, and period.

A ball on a 2 m string moves in a horizontal circle at constant speed of 3 m/s. Find the angular velocity and centripetal acceleration.

A 10 kg object explodes into a 6 kg piece moving right at 5 m/s and a 4 kg piece. Find the velocity of the 4 kg piece.

Calculate Earth's escape velocity. How fast must an object move to escape Earth's gravity?

A motor lifts 100 kg by 10 m in 5 seconds. Calculate the average power.

A constant 10 N force acts on a 2 kg box over a distance of 5 m. Find the final speed if the box starts from rest.

A ball rolls from rest down a frictionless 5 m high ramp. What is its speed at the bottom?

A 1 kg ball moving at 5 m/s collides with and sticks to a 2 kg ball at rest. Find the final velocity.

A 1 kg ball moving at 5 m/s elastically collides with a 2 kg ball at rest. Find the final velocities.

A roller coaster cart starts from rest at the top of a 20 m high hill. Find its speed at the bottom (no friction).