Question

A pendulum consists of a small object hanging from the ceiling at the end of a string of negligible mass. The string has a length of 0.75 m. With the string hanging vertically, the object is given an initial velocity of 1.9 m/s parallel to the ground and swings upward in a circular arc. Eventually, the object comes to a momentary halt at a point where the string makes an angle θ with its initial vertical orientation and then swings back downward. Find the angle θ. Muitiple-Concept Example 5 reviews many of the concepts that play a role in this problem. An extreme skier, starting from rest, coasts down a mountain that makes an angle of 26.0∘ with the horizontal. The coefficient of kinetic friction between her skis and the snow is 0.182 . She coasts for a distance of 15.8 m before coming to the edge of a cliff. Without slowing down, she skis off the cliff and lands downhill at a point whose vertical distance is 4.72 m below the edge. How fast is she going just before she lands?

Answer 1

To find the angle θ, use conservation of energy and equate the gravitational potential energy at the top of the swing to the rotational kinetic energy at the bottom of the swing.

Step-by-step explanation:

The angle θ can be found using conservation of energy. At the top of the swing, the object has gravitational potential energy which is equal to mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above the lowest point of its arc. At the bottom of the swing, all of the gravitational potential energy is converted into rotational kinetic energy.

Therefore, mgh = 1/2Iω^2, where I is the moment of inertia of the object and ω is the angular velocity of the object. The moment of inertia for a solid sphere is given by I = 2/5mr^2, where r is the radius of the sphere.

Substituting the values given in the question, we can solve for the angle θ.