Rotational, Circular, and Harmonic Motion - MCAT Chemical and Physical Foundations of Biological Systems

The key to answering this question is to recall the following important formula for a simple pendulum: .

For a pendulum, the angular velocity is given by the equation , where is the acceleration due to gravity and is the length of the pendulum. Of the available answer choices, only changing the length of the string will effect the angular velocity. does not depend on mass or the release point of the pendulum.

Choice 2 is correct; this is how a classical slingshot operates. A body in motion tends to remain in motion unless a force disturbs that motion. At any point in time, the insect is moving in a straight line along a line tangent to the circumference of the wheel. When the little creature leaves the wheel, there is no longer any force acting on it, so it will move in a straight line, not a curvilinear path. Newtonian kinetics of course includes these concepts, because they relate closely to the motion of celestial bodies.

Since the object is traveling in a circle, it is constantly changing direction. This means that the direction of the velocity is also changing, even if the magnitude is not. Because the magnitude of the velocity does not change, we can assume that acceleration is constant.

Note that we are asked the linear velocity of the pulley, not the angular velocity. With Newton’s second law, we know that F = ma. Additionally, in rotational motion, we know that . The force (F) the instant the rope is cut is due to the platform with the 4 weights. Knowing the mass and radius of the pulley, we can solve for the linear velocity:

Maximum tension occurs when the ball is at its bottommost point, as the string experiences both the downward force of gravity and centripetal force.

An object moving in constant circular motion must have centripetal acceleration, directed towards the center of its path, in order to maintain its circular motion. So and is towards the center of car A's path.

An object moving linearly at constant velocity has 0 acceleration, so .

In general, we can think of acceleration as a change in either the magnitude or direction of an object's velocity. Neither of these cars has a change in the magnitude of its velocity, but car A does experience a change in the direction of its velocity, while car B does not.

Centripetal acceleration is given by the equation:

We are given the angular velocity and the diameter. We can easily solve for the radius from the diameter, and use the angular velocity to find the linear velocity.

Convert the angular velocity to the proper units by changing revolutions to radians.

Now we can convert the angular velocity to the linear velocity using the radius.

Use the radius and the linear velocity to calculate the centripetal acceleration from the original equation.

Centripetal force is given by the equation:

The length of the string represents the radius of the circle being formed. Use the given string length as the radius, along with the mass of the ball and the velocity, to calculate the centripetal force.

To calculate g's, we need to know the centripetal force the rider is experiencing. To calculate centripetal force, we need to know how fast the ride is traveling at the top of the loop. To do this, we will use the equation for conservation of energy:

If we set the height of the top of the loop to 10m, we can say that there is no initial potential energy. Since we are solving for final velocity, let's rearrange this for final kinetic energy:

Plugging in the expressions for each of these terms, we get:

Rearranging for final velocity:

We have all of the values except for how far the coaster travels. It travels from the bottom to the top of the loop, which is half the circumference of the circle:

Now we can plug in our values and solve:

Now that we know how fast the ride is traveling at the top of the loop, we can calculate the centripetal force on the man:

Now that we know the force in Newtons, we can convert this value to g's.

For this man:

Therefore,

In this case, the kinetic energy of the spring is at a minimum. This is because, as the question indicates, the spring is not moving. At the same time, because the spring is compressed as far is it can be compressed, we know that its potential energy is at a maximum. The total energy of the spring therefore cannot be zero.

The only factor that affects the period of a pendulum is the length of the pendulum. Therefore we can ignore any of the other answers which include other factors. The equation for the period of a pendulum is:

We are asked for the period of the pendulum, or the time it takes to make one full revolution or swing.

The period of a pendulum is given by this formula:

Where and

It takes roughly 12 seconds for the pendulum to make a full swing. Note that the mass attached to the end of the pendulum is irrelevant.

Due to conservation of angular momentum, the angular momentum L must remain constant: .

Because of the child’s decreased distance from the axis of rotation, the moment of inertia of the system also decreases The child can be approximated as a point mass with moment of inertia , and the carousel’s moment of inertia remains unchanged. Moving the child toward the center of the carousel will decrease its moment of inertia.

To examine what happens to angular velocity, we need the definition of angular momentum, . Replacing this in the conservation of angular momentum equation, we see that . Since , then so that the product remains the same.

Angular momentum remains the same, moment of inertia decreases, and angular velocity increases.

We can apply the right-hand rule for angular velocity: if the fingers on the right hand curl in the direction of rotation, the thumb points in the direction of the angular velocity vector. In this case the fingers curl clockwise, so thumb points downwards.

The angular acceleration would decrease. Newton’s second law states that F = ma. The net force, however, is reduced in the internal friction scenario because the friction acts to oppose the motion created by the rope around the pulley. As a side note, the pulley would also heat up because the energy change due to friction is released as heat.

An object with low moment of inertia will acquire less rotational kinetic energy and more translational kinetic energy (moves faster down the ramp) than an object with higher moment of inertia.

Moment of inertia depends on how close to the object's center of mass its mass is concentrated. An object with more of its mass close to the center will have lower moment of inertia, and subsequently more translational kinetic energy. The solid sphere has lowest moment of inertia, then the disk, and then the hollow sphere, thus the solid sphere reaches the end first, then the disk, and finally the hollow sphere.

The center of mass is the point where we can consider all of the substance's mass to be concentrated. Finding the center of mass greatly facilitates mechanics problems, because we can perceive the force to be acting on only this one point.

Since the lead side of the cylinder is heavier than the aluminum side, we can conclude that the center of mass is located inside the lead side of the cylinder. Since the lead side has a mass of 60g, each inch of the lead side will have a mass of 15g. By going one inch into the lead side, there are 45g on each side of the point. As a result, we say that this is the location for the center of mass for the cylinder, as either side would weigh the same individually.

The equation relating velocity and radius is .

Because angular velocity can also be found using the equation , we can set the equations equal and solve for the velocity.

Since we are given the radius as and the frequency as , we can calculate the velocity. (Frequency can be found by the number of times an object travels around a full circle in one second).

Alternatively, we can use the radius to determine the circumference of the collider.

The particle travels twice around the collider every second.