A cube of aluminium measures 1 m along each side. In the centre of the cube there is a small spherical cavity with a diameter of 1 cm. When the cube is heated from 10 ◦C to 30 ◦C, what is the percentage increase in the volume of the central cavity? (The lin- ear thermal expansion coefficient of aluminium is 23 × 10−6 K−1 .) 0.14%

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.

What is the rate of change in a objects position?

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