Tuesday, 23 September 2014

Rotational Dynamics



Question 1
A motor produces a torque of 5Nm. It is used to accelerate a wheel of radius 10cm and moment of inertia 2kgm2 which is initially at rest. Calculate
a)the number of revolutions made by the wheel in the first 5s
b)the angular velocity after 5s
c)the centripetal acceleration of a point on the rim of the wheel after 5s.
Question 2
A cylindrical space-ship has a total mass of 2000 kg and a diameter of 2·5m. The space-ship is rotating with an angular velocity of 0·4rads-1. Two rockets are used to stop the rotation. See diagram below.
If each rocket provides a constant force of 50N, how long will it take to stop the rotation?
(For simplicity, consider the space-ship to be a thin-walled cylinder with all the mass concentrated in the walls.)
Question 3
Two wheels have the same mass, m. Wheel A has radius ra = 10cm and is a uniform, solid wheel. Wheel B has radius rb = 15cm and has all its mass concentrated in a thin rim (something like a bicycle wheel). Both wheels are caused to accelerate from rest by a torque TORQUE = 8Nm. Calculate
a)the ratio (K.E. of A)/(K.E. of B) after 10s of acceleration
b)
the ratio (K.E. of A)/(K.E. of B) when each wheel has completed 5 revolutions. (The wheels will, of course, take different times to complete 5 revolutions, but this has no relevance to this part of the question.)
Question 4
A uniform rod of length  and mass m has a moment of inertia I1 = 6kgm2 when rotated about an axis very near one end. What is its moment of inertia, I2, when rotating about an axis through its centre?
Question 5
A solid cylinder of mass 0·5kg is rolling (without slipping) along a horizontal surface at 3ms-1. Calculate its total kinetic energy.
Question 6

a)
State the law of conservation of angular momentum.
b)
Use this law to explain how an ice-skater can change his/her angular velocity.
c)
Halley’s comet has a very eccentric orbit around the sun; that is, the ratio, furthest distance from the sun (aphelion) to nearest distance to the sun (perihelion), is large. At aphelion, it is about 4.8 × 109km and at perihelion it is only about 8.8 × 107km from the sun. At aphelion, its speed is about 5.3 × 103ms-1. Use the principle of conservation of angular momentum to calculate its speed at perihelion.