# 2023 AP Physics 1 Free Response #3

TLDRThis video script discusses the solutions to a problem from the AP Physics 1 2023 exam. The presenter provides their best guess for the answers, assuming no mistakes are present. The problem involves a mass attached to a spring on a rotating rod, with varying speeds and spring stretches. The presenter explains the force diagrams for different scenarios, compares the tangential speeds at different times, and derives an expression for the net force on the block in circular motion, using the spring constant, stretch distance, and mass.

###### Takeaways

- π The video discusses a free response question from the AP Physics 1 2023 exam, focusing on a block attached to a spring rotating on a horizontal table.
- π The problem involves analyzing forces at two different times, T1 and T2, when the block moves in a circular path with different tangential speeds, V1 and V2, and the spring is stretched by distances D1 and D2.
- π‘ At both times, the spring force pulls the block towards the center, with the force at T2 being greater due to a larger stretch (D2 > D1).
- π The spring force is proportional to the stretch distance, represented by the equation F_spring = k * x.
- π The tangential speed V2 is greater than V1 because a larger centripetal force and acceleration are needed to maintain the faster speed.
- π The reasoning for the relative lengths of the force arrows is based on the proportionality of the spring force to the stretch distance.
- π To determine the net force on the block when the spring is stretched by distance D, the net force is given by F_net = k * D.
- π The centripetal force equation used is F_net = m * v^2 / r, where r is the total radius (D + L).
- βοΈ By solving for V, the expression for tangential speed is derived as V = sqrt((k * D * (D + L)) / m).
- β The final tangential speed expression aligns with the initial reasoning, confirming that a greater stretch (D2 > D1) results in a greater tangential speed (V2 > V1).

###### Q & A

### What is the situation described in the AP Physics 1 2023 exam scenario?

-A small block of mass m0 is attached to a spring with spring constant k0 on a horizontal table. The spring is attached to a rod that can rotate about its axis at various speeds due to a motor. Initially, the block is at rest with the spring at its unstretched length L. At different times, T1 and T2, the rod spins at different speeds, causing the block to move in a circular path with tangential speeds V1 and V2, respectively, and the spring to stretch to distances D1 and D2, with D2 being greater than D1.

### What is the significance of the spring force in the given scenario?

-The spring force is significant as it provides the centripetal force necessary for the block to move in a circular path. The magnitude of the spring force is proportional to the distance the spring is stretched from its equilibrium length, which is represented by the spring constant k0 and the stretch distance D.

### How does the spring force change when the block moves with different tangential speeds V1 and V2?

-The spring force increases when the block moves with a higher tangential speed because the spring is stretched more, resulting in a greater centripetal force needed to maintain the circular motion. Since D2 is greater than D1, the spring force at T2 is greater than at T1.

### What is the relationship between the spring force and the tangential speed of the block?

-The tangential speed of the block is directly related to the spring force. A greater spring force implies a larger net force and thus a larger centripetal acceleration, which in turn requires a higher tangential speed for the block to maintain its circular path.

### Why is the tangential speed V2 greater than V1 according to the reasoning in the script?

-The tangential speed V2 is greater than V1 because at T2, the spring is stretched more (D2 > D1), resulting in a larger spring force. This larger force corresponds to a larger centripetal acceleration, which is necessary for the block to move at a higher speed in its circular path.

### What is the expression for the net force on the block when it is traveling in a circular path with the spring stretched by distance D?

-The net force on the block, F_net, can be expressed as the product of the spring constant k0 and the stretch distance D, since the spring force is the only external force acting on the block in the radial direction.

### How is the radius of the circular path related to the spring's unstretched length L and the stretch distance D?

-The radius of the circular path, r, is the sum of the spring's unstretched length L and the stretch distance D from the equilibrium position. This is because the block moves in a circle with the center at the equilibrium position of the spring.

### What is the formula for the tangential speed V of the block when considering the net force and the radius of the circular path?

-The tangential speed V of the block can be found by taking the square root of the ratio of the net force (k0 * D) multiplied by the total radius (D + L) and divided by the mass of the block m0, i.e., V = sqrt((k0 * D * (D + L)) / m0).

### How does the provided reasoning in the script support the relationship between the spring force and the tangential speed of the block?

-The reasoning in the script supports the relationship by explaining that a larger spring force, due to a greater stretch distance, results in a larger centripetal force and acceleration, which is necessary for the block to move at a higher tangential speed in its circular path.

### What is the significance of the unstretched length L in the scenario described in the script?

-The unstretched length L of the spring is the reference point from which the stretch distance D is measured. It is crucial in determining the total radius of the circular path and in calculating the net force and tangential speed of the block.

### How can the script's explanation of the spring force and tangential speed be applied to real-world physics problems?

-The script's explanation can be applied to real-world physics problems involving circular motion, such as the tension in a cable attached to a moving object, the force exerted by a spring in a mechanical system, or the analysis of a car moving in a circular track, where understanding the relationship between force, acceleration, and speed is essential.

###### Outlines

##### π AP Physics 1 Exam Analysis

This paragraph discusses the analysis of the AP Physics 1 2023 exam's free response question. The speaker provides their best guesses for the solutions and promises to update the PDF and comments if any mistakes are found. The scenario involves a block attached to a spring on a rotating rod, with the spring's force and the block's motion being the focus. At time T1, the block moves in a circular path with a constant tangential speed V1 and the spring is stretched to distance D1. At time T2, the block's speed V2 and spring stretch D2 are greater, with D2 > D1. The speaker explains the force exerted by the spring and how it is represented in a free body diagram, emphasizing that the spring force is proportional to the stretch distance and is directed towards the center of the circular path. The reasoning for the relative magnitudes of the spring forces at T1 and T2 is based on the stretch distances D1 and D2.

##### π Centripetal Force and Tangential Speed Comparison

In this paragraph, the speaker delves into the relationship between centripetal force, acceleration, and tangential speed in the context of the block moving in a circular path. They argue that at time T2, the block's speed V2 is greater than V1 at time T1 due to the increased centripetal force resulting from the greater spring stretch (D2 > D1). The explanation is based on the concept that a larger centripetal force leads to greater centripetal acceleration, which in turn implies a higher tangential speed. The speaker also discusses the expression for the net force on the block when it travels in a circular path with the spring stretched a distance D from its unstretched length L. The net force is expressed in terms of the spring constant k0, the stretch distance D, and the block's mass m0, with the formula involving the radius of the path (D + L) and the tangential speed V. The summary concludes with a confirmation that the derived equation for the tangential speed V is consistent with the reasoning provided earlier regarding the relationship between the spring stretch and speed.

###### Mindmap

###### Keywords

##### π‘AP Physics 1

##### π‘Free response

##### π‘Spring constant

##### π‘Tangential speed

##### π‘Centripetal force

##### π‘Streched length

##### π‘Unstretched length

##### π‘Centrifugal acceleration

##### π‘Net force

##### π‘Free body diagram

##### π‘Centripetal acceleration

###### Highlights

Free response three of the AP Physics 1 2023 exam solutions are presented, with a note that any mistakes will be updated in the PDF and noted in a pinned comment.

A small block of mass m0 is attached to a spring with spring constant k0 on a horizontal table, with the rod attached to a motor for rotation.

At time T1, the rod spins with the block moving in a circular path at constant tangential speed V1 and the spring stretched by distance D1.

At time T2, the rod spins faster, the block moves in a circular path with constant tangential speed V2 and the spring stretched by greater distance D2.

The force exerted on the block by the spring is to be drawn in free body diagrams, with the spring force represented by distinct arrows.

The spring force is proportional to the stretch distance, with a greater force when the spring is stretched more at T2 compared to T1.

The block's tangential speed V1 at T1 is compared to the tangential speed V2 at T2, with reasoning provided for V2 being greater than V1.

The reasoning for V2 being greater than V1 is based on the increased centripetal force and acceleration when the spring is stretched more.

The net force exerted on the block is expressed in terms of the spring force and the centripetal acceleration formula.

The radius of the circular path is considered to be the sum of the spring stretch distance D and the unstretched length L.

An expression for the block's tangential speed V is derived, relating it to the spring constant, stretch distance, and mass.

The derived equation for tangential speed V is confirmed to be consistent with the reasoning provided in part A regarding the relationship between D and V.

The importance of considering the spring force and stretch distance in analyzing the block's motion in a circular path is emphasized.

The problem-solving approach involves understanding the relationship between spring force, centripetal force, and tangential speed.

The difficulty of the problem is acknowledged, especially the challenge of justifying the relationship between force and speed without using equations.

The significance of the block's increased speed when the spring is stretched more is highlighted, as it requires a larger centripetal force for circular motion.

The final part of the transcript involves considering a scenario where the block travels in a circular path with the spring stretched by distance D, leading to an expression for the net force.

###### Transcripts

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