A particle moving along the y-axis is in a system with potential energy U = 4y³ J, where y is in m. What is the y-component of the force on the particle at y = 0 m, 1 m, and 2 m?

Verified step by step guidance

1. Step 1: Recall the relationship between force and potential energy. The force acting on a particle in a potential energy field is given by the negative gradient of the potential energy. Mathematically, this is expressed as F_y = -dU/dy, where F_y is the force in the y-direction and U is the potential energy.
2. Step 2: Differentiate the given potential energy function U = 4y^3 with respect to y. Using the power rule for differentiation, dU/dy = d(4y^3)/dy = 12y^2.
3. Step 3: Substitute the expression for dU/dy into the force equation. This gives F_y = -12y^2.
4. Step 4: Evaluate the force at the specified positions along the y-axis. For y = 0 m, substitute y = 0 into F_y = -12y^2. Similarly, for y = 1 m and y = 2 m, substitute y = 1 and y = 2 into the same equation.
5. Step 5: Interpret the results. The force at each position indicates the direction and magnitude of the force acting on the particle due to the potential energy field. Note that the force is negative, meaning it acts in the opposite direction of increasing potential energy.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position in a force field, such as gravitational or elastic fields. In this case, the potential energy U = 4y^3 J indicates that the energy depends on the position y of the particle along the y-axis. Understanding how potential energy varies with position is crucial for analyzing the forces acting on the particle.

Force and Potential Energy Relationship

The force acting on a particle can be derived from the potential energy function using the relationship F = -dU/dy. This means that the force is equal to the negative gradient of the potential energy with respect to position. This concept is essential for determining the force at specific positions by calculating the derivative of the given potential energy function.

Calculating Derivatives

Calculating derivatives is a fundamental mathematical skill used to find rates of change. In the context of this problem, taking the derivative of the potential energy function U = 4y^3 with respect to y allows us to find the force at different positions. Mastery of differentiation techniques is necessary to accurately compute the force values at y = 0 m, 1 m, and 2 m.