Ch 08: Dynamics II: Motion in a Plane

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 08: Dynamics II: Motion in a Plane

Problem 32a

Chapter 8, Problem 32a

CALC A 100 g bead slides along a frictionless wire with the parabolic shape y = (2m^-1) x². Find an expression for a_y, the vertical component of acceleration, in terms of x, v_x, and a_x. Hint: Use the basic definitions of velocity and acceleration.

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Step 1: Start by recalling the relationship between position, velocity, and acceleration. Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. For this problem, the bead's position is given by the parabolic equation y = (2m⁻¹)x².

Step 2: Differentiate the given equation y = (2m⁻¹)x² with respect to time (t) to find the vertical velocity vᵧ. Using the chain rule, we get vᵧ = dy/dt = d/dx[(2m⁻¹)x²] * dx/dt = 4m⁻¹ * x * vₓ, where vₓ = dx/dt is the horizontal velocity.

Step 3: Differentiate vᵧ = 4m⁻¹ * x * vₓ with respect to time (t) to find the vertical acceleration aᵧ. Again, use the chain rule: aᵧ = dvᵧ/dt = d/dx[4m⁻¹ * x * vₓ] * dx/dt + d/dvₓ[4m⁻¹ * x * vₓ] * dvₓ/dt.

Step 4: Simplify the expression for aᵧ. The first term becomes (4m⁻¹ * vₓ) * vₓ = 4m⁻¹ * x * aₓ, where aₓ = dvₓ/dt is the horizontal acceleration. The second term involves the derivative of vₓ with respect to time, which contributes to the overall expression for aᵧ.

Step 5: Combine the terms to express aᵧ in terms of x, vₓ, and aₓ. The final expression for aᵧ is aᵧ = 4m⁻¹ * x * aₓ + 4m⁻¹ * vₓ². This represents the vertical component of acceleration in terms of the given variables.

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Key Concepts

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

Acceleration

Acceleration is the rate of change of velocity with respect to time. In this context, it can be broken down into components, such as the vertical component (aᵧ) and the horizontal component (aₓ). Understanding how to express acceleration in terms of position and velocity is crucial for analyzing motion along a curved path.

Velocity Components

Velocity can be decomposed into its components along different axes, typically horizontal (vₓ) and vertical (vᵧ). For a particle moving along a curve, the relationship between these components is essential for determining the overall motion. The horizontal and vertical components can be related through the geometry of the path, which in this case is a parabolic shape.

Curvilinear Motion

Curvilinear motion refers to the motion of an object along a curved path. In this scenario, the bead moves along a parabolic wire, which means its acceleration and velocity must be analyzed in relation to the curvature of the path. The shape of the trajectory affects how the components of acceleration and velocity are calculated, particularly when using calculus to relate position, velocity, and acceleration.

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CALC A 100 g bead slides along a frictionless wire with the parabolic shape y = (2m-1) x2. Find an expression for ay, the vertical component of acceleration, in terms of x, vx, and ax. Hint: Use the basic definitions of velocity and acceleration.

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Key Concepts

Acceleration

Velocity Components

Curvilinear Motion

CALC A 100 g bead slides along a frictionless wire with the parabolic shape y = (2m^-1) x². Find an expression for a_y, the vertical component of acceleration, in terms of x, v_x, and a_x. Hint: Use the basic definitions of velocity and acceleration.