A uniform solid sphere of mass M and radius R is released from rest at the top of an 8.0-m-high hill. Assume the sphere rolls without slipping. (a) What is its translational speed v at the bottom of the hill? (b) What would v be if the sphere’s mass was doubled? (c) What would v be if the sphere’s radius was doubled? (d) What would v be if the hill was only 4.0 m high? (e) What would v be if the hill was still 8.0 m high, but the sphere was hollow instead of solid?
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
A uniform solid sphere of mass M and radius R is released from rest at the top of an 8.0-m-high hill. Assume the sphere rolls without slipping.
(a) What is its translational speed v at the bottom of the hill?
(b) What would v be if the sphere’s mass was doubled?
(c) What would v be if the sphere’s radius was doubled?
(d) What would v be if the hill was only 4.0 m high?
(e) What would v be if the hill was still 8.0 m high, but the sphere was hollow instead of solid?
Hint: There’s no need to start each of these calculations from scratch. Begin by reworking the steps shown in class to find the final speed v of a rigid body rolling down an incline using conservation of energy. You should obtain an expression for v in terms of the free-fall acceleration g, the incline’s height h, and a constant c that’s determined by the object’s shape. You can then use this expression to answer each part.
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