Work and Energy Test. 4

On a banked road, the road is tilted. The normal force is perpendicular to the road surface, and its horizontal component points towards the center of the curve, providing the centripetal force.

While centripetal acceleration changes the direction, tangential acceleration (a_t = rα) acts along the tangent to the path and is responsible for changing the object's speed.

The simulated gravity is the centripetal acceleration (a_c = rω²). To achieve a_c = g, the required angular velocity (ω), and thus frequency, depends on the radius 'r' of the station.

By pulling her limbs in, the diver decreases her moment of inertia (I). To conserve angular momentum (L = Iω), her angular velocity (ω) must increase.

Similar to linear momentum, the total angular momentum of an isolated system (one with zero net external torque) is conserved.

Moment of inertia (I) is determined by the total mass of the object and how that mass is geometrically distributed relative to the axis of rotation (I = Σmr²).

Objects in orbit are continuously falling towards the central body but have enough tangential velocity to miss it. This state of constant free-fall results in the feeling of weightlessness.

The specific speed required to maintain a stable orbit at a certain altitude is called orbital velocity. For low-Earth orbit, this is about 7.9 km/s.

Angular acceleration (α) is the rate of change of angular velocity, so its unit is radians per second per second, or rad/s².

The solid sphere has a smaller moment of inertia than the hollow sphere. This means less of its potential energy is converted to rotational KE and more to translational KE, making it faster.

The kinetic energy due to rotation is given by K.E_rot = ½Iω², which is analogous to the linear kinetic energy formula K.E = ½mv².

A rolling object is both moving from one place to another (translation) and spinning (rotation), so its total kinetic energy is the sum of both forms.

The apparent weight (normal force) is N = m(g-a). When accelerating down, the apparent weight is less than the true weight.

Angular momentum (L) is the rotational equivalent of linear momentum and is calculated as the product of the moment of inertia and angular velocity.

A geostationary satellite orbits above the equator with an orbital period of exactly 24 hours, causing it to appear stationary from the ground.

Torque (τ) is the rotational equivalent of linear force; it is a twisting force that causes an object to undergo angular acceleration.

A flywheel has a large moment of inertia and is used to store rotational kinetic energy, resisting changes in speed and ensuring smoother operation.

By spinning a ring-shaped space station, the normal force exerted by the inner floor on an astronaut provides the necessary centripetal force, simulating the sensation of gravity.

Just as F=ma relates force, mass, and linear acceleration, τ = Iα relates torque (τ), moment of inertia (I), and angular acceleration (α).

Angular acceleration α = Δω/Δt = (10 - 2) rad/s / 4 s = 8/4 = 2 rad/s².