CALC The potential energy for a particle that can move along the x-axis is U = Ax2 + B sin(πx/L), where A, B, and L are constants. What is the force on the particle at (a) x = 0, (b) x = L/2, and (c) x = L?

Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 63

Chapter 10, Problem 63

CALC The potential energy for a particle that can move along the x-axis is U = Ax²+ B sin(πx/L), where A, B, and L are constants. What is the force on the particle at (a) x = 0, (b) x = L/2, and (c) x = L?

Verified step by step guidance

Step 1: Recall the relationship between force and potential energy. The force on a particle is given by the negative gradient of the potential energy function: F(x) = -dU/dx.

Step 2: Differentiate the given potential energy function U(x) = Ax² + B sin(πx/L) with respect to x. Use the chain rule for the sine term. The derivative is: dU/dx = 2Ax + B(π/L)cos(πx/L).

Step 3: For part (a), substitute x = 0 into the expression for the force: F(0) = -[2A(0) + B(π/L)cos(π(0)/L)]. Simplify the expression to find the force at x = 0.

Step 4: For part (b), substitute x = L/2 into the expression for the force: F(L/2) = -[2A(L/2) + B(π/L)cos(π(L/2)/L)]. Simplify the expression to find the force at x = L/2.

Step 5: For part (c), substitute x = L into the expression for the force: F(L) = -[2A(L) + B(π/L)cos(π(L)/L)]. Simplify the expression to find the force at x = L.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 is a function of position x, defined by the equation U=Ax²+B sin(πx/L). 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 relation F = -dU/dx. 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 along the x-axis, as it directly links the potential energy to the force experienced by the particle.

Differentiation

Differentiation is a fundamental mathematical process used to find the rate at which a quantity changes. In the context of this problem, differentiation is applied to the potential energy function U with respect to x to calculate the force. Mastery of differentiation techniques is necessary to accurately compute the force at the specified positions x=0, x=L/2, and x=L.

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