What are the formulas to find the acceleration of an object rolling without slipping on an inclined plane?

Question

BlackTom AI · Accepted Answer

Let's examine each option in turn and see what it expresses about the dynamics of a rolling object on an incline.

Option a: a = Δv/Δt and α = Δω/Δt
- This pair mirrors definitions of linear and angular acceleration from kinematics, which describe how velocity and angular velocity change over time. While these relations are always true in a kinematic sense, they do not, by themselves, represent the forces and torques acting on the system. For rolling without slipping on an incline, you must connect accelerations to the forces (gravity, normal, friction) and to the rotational dynamics via torques. So this option is incomplete as a description of the governing formulas for acceleration in this scenario.

Option b: ΣF = ma and Στ = I α
- This is the standard form of Newton's second law for both translation and rotation: the net external force equals mass times linear acceleration, and the net external torque equals the moment of inertia times angular acceleration. For an object rolling without slipping on an incline, the gravitational component along the slope provides a translational acceleration, while friction provides the torque that causes angular acceleration, connecting a and α. This pair directly establishes the dynamic equations needed to analyze the system, making it the correct set of formulas for this situation.

Option c: K = ½ mv^2 and Krot = ½ I ω^2
- These are energy expressions: kinetic energy of translation and rotational kinetic energy. They are useful for energy methods, not for writing the equations of motion in terms of forces and torques. They do not by themselves provide the accelerations nor the relationships between forces, torques, and motions on the incline, so they are not the appropriate primary formulas for acceleration analysis.

Option d: p = mv and L = I ω
- Here p represents linear momentum and L represents angular momentum. While these are fundamental quantities, they are not the acceleration equations or the direct equations of motion for a rolling object on an incline. They describe conserved or exchanged quantities in dynamics but do not specify how acceleration arises from forces and torques in this context.

In summary, option b explicitly ties together the translational and rotational dynamics through force and torque balances, which is essential for determining the acceleration of a rolling object on an incline. The other options either express kinematic definitions, energy relations, or momentum concepts that do not alone provide the governing acceleration equations for this rolling scenario.

What are the formulas to find the acceleration of an object rolling without slipping on an inclined plane?单项选择题

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