Assume that the box and disk each have mass m, the top of the incline is at height h, and the angle between the incline and the ground is θ (i.e., the incline is at an angle θ above the horizontal). Also, let the radius of the disk be R. How much sooner does the box reach the bottom of the incline than the disk? Express your answer in terms of some or all of the variables m, h, θ, and R, as well as the acceleration due to gravity g.

The lift force on an airplane of mass 897 kg  with speed v is given by c v2 (N), where c is some constant depending on the air density and wing geometry.  The lift force points perpendicular to the wings of the plane, so that if the plane banks by, say 10 degrees, then the lift force turns 10 degrees from the vertical.  Now suppose that the pilot of this plane wishes to complete a flat turn of radius 931 m with speed 104 m/s.  How much should they bank the plane, in degrees? (Please answer to the fourth decimal place - i.e 14.3225)

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A particle of mass m = 4.56 kg is in uniform circular motion around a fixed point O, with a radius of orbit r = 3.63 m. The particle experiences a net force of magnitude F = Br towards the point O, where B= 2.75 Nm-¹. What is the period of the particle's orbit? Give your answer by entering a number, specified to an appropriate number of significant figures, in the empty box below. period of orbit = m x² I

Two concentric spherical shells with uniformly distributed masses M, and M, are situated as shown in the figure below. a M1 M2 Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at the following radial distances. (Use any variable or symbol stated above along with the following as necessary: a, b, c, and G for the gravitational constant.) (a) r = a F = (b) r = b F = (c) r = c F =

A skier starts from rest and slides down a slope of length L = 1080 m and angle a = 12° relative to the ground which is latitude line of λ = 57° relative to the equator. W X 2 N X L Z α Find the deflection (in meters, including sign) of the skier when it reaches the bottom of the track due to Coriolis force. Note: 1. Assume that the gravitational force is directed into the center of the Earth and it includes the centrifugal force. 2. Think about the trajectory of the skier without the effect of Coriolis force, and from there find the effect of Coriolis on the acceleration. The acceleration is time dependent and from there you can find the deflection. 3. The deflection is very small, so be very accurate with your calculation. Use g = 9.8 m/s².

Two concentric spherical shells with uniformly distributed masses M₁ and M₂ are situated as shown in the figure. Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at each of the radial distances shown in the figure. Fa NOTE: Give your answer in terms of the variables given and G when applicable (a) What is the magnitude of the net gravitational force if the particle is located outside both shells with a radial distance a? F = M₁ (b) What is the magnitude of the net gravitational force if the particle is located between the two shells with a radial distance b? Fc M₂ ****** - a (c) What is the magnitude of the net gravitational force if the particle is located inside both shells with a radial distance c? =

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Verify Kepler’s Laws of Planetary Motion. Assume that each planet moves in an orbit given by the vectorvalued function r. Let r = || r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet.