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Problem Number 1Prove that if the gravitational field strength at distance r from the center of a planet of mass M is G M / r^2, the work required to move a mass m from a point at distance r1 to a point at distance r2 from the planet with no net change in velocity is G M m ( 1/r1 - 1/r2). Derive the expression for the velocity of an object in a circular orbit at distance r from the center of the planet. Use this result to show that the KE change between circular orbits has half the magnitude of the PE change between those orbits.
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Problem Number 2Derive the expression for v(r), the velocity of a satellite orbiting at distance r from a planet of mass M. Find dv / dr.
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Problem Number 3Sketch and label force diagrams for each of the following situations:
A mass of 45 grams is attached to a cart of mass 225 grams and suspended over a pulley of negligible mass and friction. The cart is placed on a ramp whose slope is just enough to compensate for the small frictional force acting on the cart. When the system is released, what will be the acceleration of the cart?
Answer the same question if the cart is placed on a ramp making an angle of 4 degrees with horizontal, with the cart being pulled down the ramp, and if the frictional force is .016 times the normal force on the cart.
At what ramp angle will the cart experience zero acceleration?
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Problem Number 4What is the centripetal acceleration of a satellite orbiting at a radius of 14900 km from the center a certain planet if it is moving at 17600 m/s in that orbit? What is its orbital period (i.e., how long does it take to complete an orbit)?
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Problem Number 5What is the initial launch speed of a basketball player whose vertical leap is 84 cm? code `t
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Problem Number 6If one Calorie of food energy is 1000 calories (note large and small c), and if one calorie is about 4.19 Joules of energy, and if your body is capable of converting about 15% of the food energy you consume into useful work, then if you used up all the energy in a 1160-Calorie meal to power a special machine which would convert all your work into your kinetic energy, what speed could you attain (you may assume your own mass or, if you prefer, assume a mass of 86 Kg).
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Problem Number 7A simple harmonic oscillator with mass 1.22 kg and restoring force constant 360 N/m is released from rest at a displacement of .43 meters from its equilibrium position.
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Problem Number 8Explain why the work required to stretch a spring or other elastic object with a linear restoring force, of form F = - kx, from its equilibrium position to displacement x is `dW = .5 k x^2, and why we hence say that this is the elastic potential energy of the object in this position.
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Problem Number 9Derive the expression for v(r), the velocity of a satellite orbiting at distance r from a planet of mass M. Find dv / dr.
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Problem Number 10A Ferris wheel with radius 21 meters is moving fast enough that at the top of its arc a 150-lb person at the rim of the wheel has an apparent weight of only 99 pounds.
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Problem Number 11If an automobile of mass 531 kg is moving around a circular track 68.99 meters in diameter with angular velocity 1.08 rad/sec, then how fast is it moving and what centripetal force is acting on it?
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Problem Number 12A simple harmonic oscillator is subjected to a net restoring force F = - 40 N/m * x at displacement x from equilibrium. It is observed to undergo 47 complete cycles of motion in 48 seconds. What is its mass?
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Problem Number 13What is the equation of motion of pendulum of length L which is given an initial velocity v0 at position x0 at clock time t = 0? What are the expressions for its velocity and acceleration functions v(t) and a(t)?
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Problem Number 14Give an example of a situation in which you are given a,
Ds and Dt, and reason out all possible conclusions that could be drawn from these three quantities, assuming uniform acceleration. Accompany your explanation with graphs and flow diagrams. Show how to generalize your result to obtain the symbolic expressions for v0 and vf..
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Problem Number 15What is the effective force constant for a simple pendulum of mass .2 kg and length 2.39 meters? What therefore is its period of oscillation?