Circular Motion Test. 4

This arrangement produces a pure torque (turning effect) without any net translational force.

By the perpendicular axis theorem, I_z = I_x + I_y. For a disk, I_x = I_y and I_z = (1/2)MR². So, 2I_x = (1/2)MR², which gives I_x = (1/4)MR².

Using τ = Iα, we get α = τ/I = 10 N·m / 2 kg·m² = 5 rad/s².

This condition ensures that the point of contact with the surface is instantaneously at rest.

First, ωᵢ = 1200 × (2π/60) = 40π rad/s. Then, α = (ω_f - ωᵢ)/t = (0 - 40π)/10 = -4π rad/s². The magnitude is 4π rad/s².

As the radius increases, the moment of inertia (I ∝ R²) increases. Since angular momentum (L = Iω) is conserved, the angular velocity (ω) must decrease.

The period (T) is 24 hours = 86400 s. Frequency is f = 1/T, so f = 1/86400 Hz.

Angular momentum is L = r × p. If the particle moves along a line passing through the reference point, r is parallel to p, so their cross product L is zero and conserved.

This is the rotational equivalent of the linear power formula P = Fv.

This is the rotational work formula, analogous to linear work W = Fd.

From F = G(m₁m₂/r²), we rearrange to get G = Fr²/(m₁m₂). The dimensions are [MLT⁻²][L²]/[M²] = [M⁻¹L³T⁻²].

In general relativity, massive objects warp spacetime, and this curvature is what we perceive as gravity.

Using θ = ωᵢt + (1/2)αt², with ωᵢ = 0, we get θ = (1/2)(2)(4)² = 16 rad.

Using the right-hand rule, if you curl your fingers counter-clockwise, your thumb points out of the page.

The original acceleration is a = rω². The new acceleration is a' = (r/2)(2ω)² = (r/2)(4ω²) = 2(rω²) = 2a.

To maintain a stable orbit at a greater distance, less orbital speed is required to balance the weaker gravitational pull.

This is the definition of the cross product τ = r × F. The direction is given by the right-hand rule.

At the top, the minimum centripetal force is provided by gravity alone (tension T=0). So, mg = mv²/R, which gives v = √(gR).

Just as linear impulse equals the change in linear momentum, angular impulse (τΔt) equals the change in angular momentum (ΔL).

Planck's constant has units of Joule-seconds (J·s), which gives dimensions of [ML²T⁻²]·[T] = [ML²T⁻¹], the same as angular momentum.