Question

Which of the following are in static equilibrium? a) a pendulum at the top of its swing b) a merry-go-round spinning at constant angular velocity c) a projectile at the top of its trajectory (with zero velocity) d) all of the above e) none of the above

Short answer

a) A pendulum at the top of its swing b) A merry-go-round spinning at constant angular velocity c) A projectile at the top of its trajectory (with zero velocity) d) All of the above e) None of the above Answer: e) None of the above Explanation: In each of the given situations (a, b, and c), the objects are either under the influence of a net force (gravity, for example) or not at rest, which means they do not meet the conditions for static equilibrium. Therefore, none of the given scenarios are in static equilibrium.

Step-by-step solution

Situation A: A pendulum at the top of its swing

In this case, the pendulum is at the highest point in its swing, and its velocity is momentarily zero. However, the pendulum is still under the influence of gravitational force, which causes it to accelerate back towards its equilibrium point. Since there is still a net force acting on the pendulum (gravity), it is not in static equilibrium.

Situation B: A merry-go-round spinning at constant angular velocity

A merry-go-round spinning at a constant angular velocity has a balanced net torque, as the rotational forces are equal and opposite, and the net force is zero. However, since the merry-go-round is in motion and not at rest, it is not in static equilibrium. It is, instead, in dynamic equilibrium.

Situation C: A projectile at the top of its trajectory (with zero velocity)

Although the projectile has reached a point where its velocity is momentarily zero, it is still under the influence of gravity, which causes it to accelerate back downwards. This means there is a net force acting on the projectile (gravity). Therefore, it is not in static equilibrium.

Final Answer

Out of the given scenarios A, B, and C, none of them meet the conditions for static equilibrium. Therefore, the correct answer is: e) none of the above

Key concepts

Dynamic Equilibrium

When studying physics, dynamic equilibrium is a crucial concept. It refers to a situation where an object is moving at a constant velocity while the applied forces are balanced. This means that, even though the object is not at rest, there is no net change in its motion.

Take the example of a merry-go-round spinning at a constant angular velocity: it's a perfect illustration of dynamic equilibrium. Despite its continuous motion, there's a balance between the centrifugal force pushing outward and the centripetal force pulling toward the center, creating a steady state where the merry-go-round rotates without speeding up or slowing down. It's important to remember here, unlike static equilibrium, dynamic equilibrium can occur when an object is in motion, as long as that motion is uniform.

Characteristics of Dynamic Equilibrium

• Net force is zero, but the object is in motion.
• Net torque is balanced, leading to constant rotational speed if it's a rotating object.
• There is no acceleration, indicating a constant velocity.

Trying to move the merry-go-round faster or slow it down requires a force to disrupt this delicate state of balance. This is an essential distinction from static equilibrium, where the object remains at rest.

Gravity

Gravity is the force that attracts two bodies towards each other. On Earth, this force is what gives us weight and causes objects to fall towards the ground when dropped. It's a pervasive influence on all physical objects, and it's essential when discussing static and dynamic equilibriums.

Using the scenarios from the exercise, gravity is the force acting on both the pendulum at the top of its swing and the projectile at the top of its trajectory. It's what pulls them back towards their equilibrium points — the lowest point in the swing for the pendulum and the ground for the projectile. Despite these objects having zero velocity at the top of their motion, they are not in static equilibrium because gravity acts to accelerate them downward, creating a net force.

Impact of Gravity on Equilibrium

• Gravity is a constant force pulling objects toward the center of the planet.
• An object at rest could still be experiencing a gravitational force, thereby not being in static equilibrium if it's able to move.
• In a dynamic system, gravity is one of the forces that must be balanced for the system to remain in equilibrium.

Understanding the role of gravity is paramount in physics as it is omnipresent and affects everything from celestial bodies to the simplest pendulum swinging from a string.

Angular Velocity

Angular velocity is a measure of how fast something is rotating. It's stated in terms of angle per unit time, usually degrees per second or radians per second. In the context of equilibrium, particularly of rotating bodies like a merry-go-round, angular velocity plays a significant role.

In the scenario where a merry-go-round spins at a constant angular velocity, it means each point on the merry-go-round is moving through the same angle in the same amount of time consistently. This consistent rotation demonstrates the dynamic equilibrium of the system, as the forces acting on the merry-go-round have been balanced to sustain this constant speed.

Key Aspects of Angular Velocity in Equilibrium

• It quantifies the rate of rotation, providing insight into how quickly an object is turning about an axis.
• For an object in dynamic equilibrium, its angular velocity is constant.
• A change in the angular velocity would indicate a net torque acting on the object, disrupting its equilibrium.

To summarize, in dynamic equilibrium, angular velocity remains unchanged, which implies that the sum of torques (rotational forces) must be zero, much like the net force in linear dynamic equilibrium.