Question

A particle moves in the xy-plane with coordinates given by x = a cos and y = a sin , where a = 1.5 meters and = 2.0 radians per second. what is the magnitude of the particle's acceleration?

Answer 1

To find the magnitude of the particle's acceleration in the given circular motion, differentiate the motion equations twice with respect to time and calculate the magnitude of the acceleration vector using the x and y components.

Step-by-step explanation:

The particle's motion is given by the equations x = a cos(ωt) and y = a sin(ωt), where a = 1.5 meters and ω = 2.0 radians per second. To find the magnitude of the particle's acceleration, we can differentiate the equations twice with respect to time. The acceleration vector can be found using the equations:

ax = -aω² cos(ωt), ay = -aω² sin(ωt)

The magnitude of the particle's acceleration can be found by taking the square root of the sum of the squares of the x and y components of the acceleration: a = √(ax² + ay²)

Plugging in the values a = 1.5, ω = 2.0, and t = 0, we can calculate the magnitude of the particle's acceleration.

Answer 2

The magnitude of the particle's acceleration is 6.0 m/s², calculated using the centripetal acceleration formula a_c = ω² * r for a particle in uniform circular motion.

Step-by-step explanation:

A particle moves in the xy-plane with coordinates given by x = a cos(ϴ) and y = a sin(ϴ), where a = 1.5 meters and ϴ = 2.0 radians per second. To find the magnitude of the particle's acceleration, one must calculate the second derivative of the position with respect to time to obtain the acceleration in the x and y components, and then find the magnitude of the resulting acceleration vector.

Assuming a uniform circular motion, the acceleration will have only the centripetal component, since there is no tangential acceleration. The centripetal acceleration (ac) can be found using the formula ac = ω2 * r, where ω is the angular velocity and r is the radius of the motion. Given that ω = 2.0 rad/s and r = 1.5 m, the magnitude of the centripetal acceleration is ac = 2.02 * 1.5, which equals 6.0 m/s2.