Other Circular and Rotational Motion Concepts - AP Physics 1

This question deals with inertia. You can solve this mathematically, but the answer can also be found conceptually.

Inertia (in physics) is the object's ability to resist change in motion. In this case, it is based on mass, and how the mass of the object is distributed around it's rotational axis (Center of mass = middle of sphere). The larger the inertia, the more resistant to rotation; the more resistant to rotation, the slower the object will move down the incline.

Have you ever seen figure skating? When skaters want to make very tight turns or rotations, they pull their arms and head really close into their chest and crouch down into a smaller formation.

Why do skaters do this? Physics of course! The closer the mass of an object is to its COM, the smaller inertia they have, and the smaller resistance to change in motion. If the same skater (same height, weight, dimensions) attempts a turn in a further stretched out position (limbs untucked, etc), they will have a larger inertia, and therefore would not be able to turn as quickly as they could when tucked.

Okay so back to the spheres. As with the skaters, the further a mass is distributed from its COM, the larger the inertia. So between spheres A and B, which do you think has a lower inertia?

Sphere A has it's mass more evenly distributed throughout itself, whereas sphere B has the same mass, but with it's weight distributed solely to the exterior of its shell. Therefore, sphere A is a better choice that sphere B.

Now, what about sphere C? Typically, the larger the mass, the larger the inertia, same scenario for the objects' radii. The larger the M and R, the larger the inertia and the more resistant the object is to rotation. The best answer for Michelle to win is object A only. If you want to plug in some numbers to the inertia equations, I'll provide them here:

In order to find the distance the wheel travels, we need a way to convert angular displacement to linear displacement. We know that the circumference of a circle (or a wheel, in this case) is . This means that in one rotation, the wheel would travel a distance equal to its circumference.

Distance of one rotation equals: .

Since the wheel travels rotations, the total distance that the wheel travels will be equal to the distance traveled by one rotation multiplied by the number of rotations.

Distance of rotations equals:

The equation for centripetal force on an object moving at a constant speed in a circular path is

The weight of the person is given to be . Assuming , we can find the mass of the person using the equation

We are given the velocity, , and the radius, . So, we can now solve for the centripetal force:

The faster that the object is moving while being spun, the greater the force of tension in the rope will be. Therefore, if we want to find the minimum speed of the bucket at the highest point in the circle such that the rope doesn't go slack, we should assume that the tension at that point is zero.

If the tension is zero, then the only force acting on the bucket at that point would be the force of gravity pulling down on the bucket. If the rope does not go slack, then the bucket must still be moving in a circular motion, so we can use the equation for circular/centripetal motion:

Since gravity is the only force, then

We can divide by on both sides to cancel that out, and then solve for :

The radius was given to us as 55 cm. We have to convert this to meters. Since , we should divide by 100 to get . Now, plug that into the equation above and solve for velocity:

The formula for centripetal acceleration is:

First, we need to solve for velocity by taking the distance Bob travels and dividing by the time it takes to travel that distance. The distance he travels is the circumference of the circle, and the time is 30 seconds, because the merry-go-round completes two revolutions per minute, or one revolution every thirty seconds:

Therefore,

Displacement is the change in distance from start to finish

Where is her position on the circular track. Since Laura starts and ends at the same place, the displacement is 0. It doesn't matter what we label her start/end position as, it's her initial position compared to her final.

The bucket would proceed in the same direction as the tangential velocity at that given time. As seen in the figure below, each position of the bucket around the circle will give rise to a new tangential velocity vector. If the bucket is going counter clockwise and is released at the "3 o'clock" position, the will be in the upward direction.

Inertia is the ability for an object to resist motion. In this case, inertia is determined by how far away the mass (weights) are distributed from the dumbbell's COM (center of handle). While there is no definitive equation for the "inertia of a dumbbell", we can conceptualize this. All inertia equations deal with a mass and a radius. The larger either of these numbers are, the larger the inertia! We know that the larger the inertia, the more resistant that object is to change in motion (or the more awkward it would be to pick up that object). With that said, it would be harder for Steve to pick up the new weight. Not because of the weight itself, but because of how that same weight is distributed around that object. Think about it this way; a piece of paper weighs roughly .16 oz. if I gave you a 1 cm cube that weighed .16 oz, which would be easier to pick up and balance on your head? I would most likely assume that you would say the cube. This is because the weight of the cube is more closely positioned near the COM, rather than the thin sheet of paper with it's weight evenly spread throughout 8.5" x 11".

The ring has more of it's mass further from the center than the disk, which is more spread out. Moment of inertia increases if there is a larger "circle" that the mass has to make. Thus, if they have the same total mass, the disk will have a lower moment of inertia, thus it will reach the bottom faster. An example of this phenomenon is that of an ice skater holding in his/her arms and legs close to the center of rotation when doing jumping spins. If the ice skater were to spread his/her limbs out, their moment of inertia increases, and they would not be able to spin as fast.

For this question, we have to consider the case of the earth orbiting the sun. Just like any scenario in which an object is moving in a circular motion, we know that there is a centripetal force that is constantly pulling the object inward. Thus, even though the object's speed may be constant, its velocity is constantly changing. This is because speed is a scalar that is not affected by direction. Velocity, on the other hand, is a vector which depends on direction.

To answer this question, we'll need to first identify all the forces acting on the hanging mass. To do this, we have to consider the forces acting along the y-axis and x-axis.

First, let's consider the forces in the y-direction. Since the mass is not moving in the y-direction, we know that the net force acting on the mass in the vertical direction is zero. We know that the weight of the mass acts downward, and the vertical component of the string's tension acts upward.

Next, let's look at the forces acting on the mass in the horizontal direction. Since the mass is moving in a uniform circular motion, we know that there is a centripetal force associated with this. Moreover, we know that the x-direction of the tension in the string is acting in the x-direction. Thus, this component of the string's tension must be providing the centripetal force.

Now, if we divide these two equations, we can cancel out the terms for tension and mass.

Further rearrangement of the above expression allows us to solve for the angle.

For this problem the ball picks up rotational velocity as well as linear velocity. Keeping this in mind we first set up our equation of conservation of energy

Because the object starts at rest the initial kinetic energy is 0 and likewise the final potential energy is 0 because it has spent all its potential energy to become kinetic.

Where m is mass of the ball, g is gravity, h is height of the ramp, v is final velocity of the ball, I is the moment of inertia of the ball, and w is the final rotational velocity of the ball.

Now because the ball is a solid sphere we can represent I and w as follows

and

Plugging this into our equation of energy conservation

Now solve for velocity:

Now plugging in our value for height, we arrive at a final answer of

For this question, we're asked to evaluate a true statement regarding circular motion. Hence, we'll need to look at each answer choice individually and consider whether it is true.

• The centripetal acceleration provided by the Earth's gravitational field is constant

This is not a true statement. Remember that acceleration is a vector. Even though the astronaut's centripetal acceleration is constant in magnitude, it is constantly changing direction as the force will always point towards the center of the Earth.

• If the astronaut's speed is doubled, his centripetal acceleration will also double

This statement is false. Remember the expression for centripetal force.

Hence, if the astronaut's speed becomes twice as great, then his acceleration will become four times as great.

• If the distance between the astronaut and the Earth were to double, the centripetal acceleration of the astronaut would also double

This statement is not true. Again, we need only look to the expression shown above for centripetal force to see why. Because the centripetal acceleration and distance are both inversely proportional to one another, doubling the distance of the astronaut from the Earth would cause the acceleration to become half as great. Thus, it would decrease, not increase.

• The astronaut is constantly in a state of free-fall

This is a true statement, and thus it is the correct answer. Objects that are orbiting the Earth experience a centripetal force due to gravity. If we picture a snapshot of the movement of the astronaut, he will be moving tangentially to his circular displacement. As the astronaut moves tangentially, the force of gravity serves as the centripetal force that "pulls" him back, such that the astronaut does not move down the tangential path. This continuous "pull" away from the tangential path at each instant essentially means that our astronaut will constantly fall towards the earth while maintaining his circular path, and is thus constantly in a state of free-fall, where he will be weightless.

If the cars are racing side by side on a circular track, then they have the same angular velocity, because they complete their circles in the same amount of time. However, the car on the outside has a larger circle to complete, and thus must have a higher linear velocity.

This question deals with inertia. You can solve this mathematically, but the answer can also be found conceptually.

Inertia (in physics) is the object's ability to resist change in motion. In this case, it is based on mass, and how the mass of the object is distributed around it's rotational axis (Center of mass = middle of sphere). The larger the inertia, the more resistant to rotation; the more resistant to rotation, the slower the object will move down the incline.

Have you ever seen figure skating? When skaters want to make very tight turns or rotations, they pull their arms and head really close into their chest and crouch down into a smaller formation.

Why do skaters do this? Physics of course! The closer the mass of an object is to its COM, the smaller inertia they have, and the smaller resistance to change in motion. If the same skater (same height, weight, dimensions) attempts a turn in a further stretched out position (limbs untucked, etc), they will have a larger inertia, and therefore would not be able to turn as quickly as they could when tucked.

Okay so back to the spheres. As with the skaters, the further a mass is distributed from its COM, the larger the inertia. So between spheres A and B, which do you think has a lower inertia?

Sphere A has it's mass more evenly distributed throughout itself, whereas sphere B has the same mass, but with it's weight distributed solely to the exterior of its shell. Therefore, sphere A is a better choice that sphere B.

Now, what about sphere C? Typically, the larger the mass, the larger the inertia, same scenario for the objects' radii. The larger the M and R, the larger the inertia and the more resistant the object is to rotation. The best answer for Michelle to win is object A only. If you want to plug in some numbers to the inertia equations, I'll provide them here:

In order to find the distance the wheel travels, we need a way to convert angular displacement to linear displacement. We know that the circumference of a circle (or a wheel, in this case) is . This means that in one rotation, the wheel would travel a distance equal to its circumference.

Distance of one rotation equals: .

Since the wheel travels rotations, the total distance that the wheel travels will be equal to the distance traveled by one rotation multiplied by the number of rotations.

Distance of rotations equals:

The equation for centripetal force on an object moving at a constant speed in a circular path is

The weight of the person is given to be . Assuming , we can find the mass of the person using the equation

We are given the velocity, , and the radius, . So, we can now solve for the centripetal force:

The faster that the object is moving while being spun, the greater the force of tension in the rope will be. Therefore, if we want to find the minimum speed of the bucket at the highest point in the circle such that the rope doesn't go slack, we should assume that the tension at that point is zero.

If the tension is zero, then the only force acting on the bucket at that point would be the force of gravity pulling down on the bucket. If the rope does not go slack, then the bucket must still be moving in a circular motion, so we can use the equation for circular/centripetal motion:

Since gravity is the only force, then

We can divide by on both sides to cancel that out, and then solve for :

The radius was given to us as 55 cm. We have to convert this to meters. Since , we should divide by 100 to get . Now, plug that into the equation above and solve for velocity:

The formula for centripetal acceleration is:

First, we need to solve for velocity by taking the distance Bob travels and dividing by the time it takes to travel that distance. The distance he travels is the circumference of the circle, and the time is 30 seconds, because the merry-go-round completes two revolutions per minute, or one revolution every thirty seconds:

Therefore,

Displacement is the change in distance from start to finish

Where is her position on the circular track. Since Laura starts and ends at the same place, the displacement is 0. It doesn't matter what we label her start/end position as, it's her initial position compared to her final.