Question

Part I

1. Here the angular acceleration is variable. What are the appropriate equations of motion? What is their lowest possible order? How are the tangential and radial acceleration related for a point on a rotating body?
2. Can a single force applied to a body change both its translational and rotational motion? Explain and provide two different examples.
3. If all planets had the same average density, how would the acceleration due to gravity at the surface of a planet depend on its radius?
4. Characterize the potential and kinetic energy balance in the cycle of motion of a simple pendulum and compare it with a physical pendulum.
5. Describe static and dynamic methods of determining the force constant of a spring.
6. Including appropriate equations, describe what a natural resonant frequency is.
7. Including appropriate equations, describe what a resonant forcing is and what are the possible consequences of a resonant forcing

Part II
1. Centrifuge. An advertisement claims that a centrifuge takes up only 0.150 m of bench space but can produce a radial acceleration of 3000g at 7500 RPM. Calculate the required radius of the centrifuge. Is the claim realistic?
2. Pressure. The 36-inch by 80-inch door of an isolation room in a hospital has an airtight but frictionless fit in its frame. The air pressure in the room is by 1% lower than the 1 atm air outside. What is the minimum force one must use to open this door?
3. Conservation of Angular momentum. Under some circumstances, a star can collapse into an extremely dense neutron star. The density of a neutron star is roughly 14 orders of magnitude greater than that of ordinary solid matter. Suppose the star is a uniform, solid, rigid sphere, both before and after the collapse. The star’s initial radius was 7x10^5 m, (comparable to the Sun); its final radius is 16 km. If the original star rotated once in 30 days, find the angular speed of the neutron star.
4. Work and power in Circular Motion. A hollow, thin-walled sphere of mass 12.0 kg and diameter 0.50 m is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by t) = At 2 + Bt 4, where the magnitude of A =1.50 and B has numerical value 1.10. (a) What are the units of the constants A and B? (b) for time t=4s find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.
5. SHM. When a 1 N weight hangs from an end of a long spring of force constant 1.5 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let this mass swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. What is the equilibrium length of the spring without the weight attached?
6. SHM. A 11.5 kg water in a 1.00-kg bucket is hanging from a vertical ideal spring of force constant 125N/m and oscillating up and down with an amplitude of 5.00 cm. When the bucket springs a leak in the bottom water drops out at a steady rate of 2.00 g/s. Is the period getting longer or shorter? When the bucket is half-full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. (c) What is the shortest period this system can have?
7. Combination Problem. On the planet Casino, a simple pendulum having a bob with mass 1.25 g and a length of 18.5 cm takes 1.42 s, when released from rest, to swing through an angle of 7.5 degrees, where it again has zero speed. The circumference of Casino is measured to be 51,400 km. What is the mass of the planet Casino? List all the assumptions necessary to arrive at your answer.

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