Classical Mechanics Answers

Earth's gravitational field:

1) Describe the gravitational field as it: (include diagram to aid answer)
a) exists within the room within you're sitting
b) as it exists around a planet such as the Earth

2) Outline how the Law of Universal Gravitation can be used to calculate the acceleration due to gravity at the Earth's surface

3) The gravitational field vector g has an average value on the surface of Earth of 9.8Nk-1 or ms-2. Show that the two alternative units quoted are equivalent

A car of mass 1000kg travels around a banked track of radius 200m. The angle of banking is 15 degrees and the speed of the car is such that frictional force is zero. By resolving force components into the vertical and horizontal directions:

a) calculate the value of N, the normal contact force
b) calculate the speed at which the car is travelling

The answers are below but i couldn't figure out to get to them.
a) 9.5x10^3 N
b) 22ms-1

A conical pendulum of mass 3.1kg, string length 3m and at an angle of 20 degrees is set up. Calculate:
a) the radius of the circle traced
b) the tension of the string
c) the period of the motion

The total energy of a particle executing simple harmonic motion of period 2\π s is 0.256J. The displacement of the particle at π/4 is 8\sqrt{2} cm. Calculate the mass of the particle.
a. 20 Kg
b. 50 Kg
c. 15 Kg
d. 30 Kg

The displacement of a particle executing simple harmonic motion is given by x=0.01cos4π(t+0.0625)m. Deduce the period of the motion.
a.1.0 s
b. 0.1 s
c. 2.5 s
d. 0.5 s

The equation of a simple harmonic motion is given as d^2x/dt^2 + ω^2x =0, where the symbols have their usual meaning. The dimension of the quantity ω^2 is
a. L^−1
b. M
c. T^−2
d. LT^−2

Calculate the average value of the potential energy over one complete cycle of a simple harmonic oscillator with stiffness k=200 N/m and amplitude a=5 cm
a. 0.125 J
b .1.500 J
c. 0.250 J
d. 0.500J

Prove the integral using the physical concept ( Or express the integral using the physical concept).

A block of wood with a cross - sectional area A and mass density ρ when it floats in the water, displaces the volume equal to a V of water. If a person (mass of person = m jumps on this piece, find a period of small oscillations.

A river, 400m wide is flowing with a rate of 2 m/s. A boat is sailing at a velocity of 10m/s with respect to water, in a direction perpendicular to the river. The time taken by boat to reach the opposite bank is?