A particle of mass m uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius R:


r→ = î R cos θ + ĵ R sin θ


with θ = ω₀t + (1/2)αt² , where the constants ω₀ and α are the initial angular velocity and angular acceleration, respectively. Determine the object’s tangential acceleration a→_tan and determine the torque acting on the object using 

(b) τ→ = Iα→ .

Verified step by step guidance

1. First, understand the given position vector r→ = î R cos θ + ĵ R sin θ, where θ = ω₀t + (1/2)αt². This equation describes the circular motion of the particle in terms of its angular position θ, which changes over time due to the initial angular velocity ω₀ and angular acceleration α.
2. To find the tangential acceleration a→_tan, differentiate the position vector r→ twice with respect to time t to get the acceleration vector a→. The differentiation should be done using the chain rule, considering that θ is a function of time.
3. After differentiating, isolate the component of the acceleration vector a→ that is tangential to the circle. This component can be found by projecting a→ onto the direction of the tangent to the circle at the particle's position, which is perpendicular to the radius vector at any instant.
4. To determine the torque τ→ acting on the object, use the formula τ→ = Iα→, where I is the moment of inertia of the particle about the axis of rotation, and α→ is the angular acceleration vector. Since the motion is in a plane, α→ can be treated as a scalar α in the direction perpendicular to the plane of motion.
5. Calculate the moment of inertia I for the particle, which for a point mass at a distance R from the axis of rotation is given by I = mR². Substitute this and the angular acceleration α into the torque formula to find τ→.

Key Concepts

Tangential Acceleration

Tangential acceleration refers to the rate of change of the linear velocity of an object moving along a circular path. It is directed along the tangent to the circle at the object's position and is calculated as the product of the angular acceleration (α) and the radius (R) of the circle. In this context, it helps determine how quickly the particle's speed is changing as it accelerates around the circle.

Torque

Torque is a measure of the rotational force applied to an object, causing it to rotate about an axis. It is calculated using the formula τ = Iα, where τ is torque, I is the moment of inertia, and α is the angular acceleration. Understanding torque is essential for analyzing the rotational dynamics of the particle as it moves along the circular path.

Moment of Inertia

The moment of inertia is a scalar value that quantifies how mass is distributed relative to an axis of rotation. It plays a crucial role in rotational dynamics, influencing how much torque is needed to achieve a certain angular acceleration. For a particle moving in a circle, the moment of inertia depends on its mass and the square of the radius of the circular path.