The position vector of a particle is given by 𝐫= 8 t 𝐢+1.2 t^2 𝐣-0.5(t^3-1) 𝐤, where t is the ti

step 1 Question we have position vector R vector equals to 8 t i cap plus 1 .2 t square j cap plus minu

The position vector of a particle is given by 𝐫= 8 t 𝐢+1.2...

The position vector of a particle is given by $\mathbf{r}=$ $8 t \mathbf{i}+1.2 t^{2} \mathbf{j}-0.5\left(t^{3}-1\right) \mathbf{k},$ where $t$ is the time in seconds from the start of the motion and where $\mathbf{r}$ is expressed in meters. For the condition when $t=$ 4 s, determine the power $P$ developed by the force $\mathbf{F}=40 \mathbf{i}-20 \mathbf{j}-36 \mathbf{k} \mathrm{N}$ which acts on the particle.

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Key Concepts

Position Vector

The position vector represents the location of a particle in space relative to a chosen origin. In kinematics, it is often expressed as a function of time and describes how the particle’s position evolves as time progresses.

Velocity Vector

The velocity vector is the derivative of the position vector with respect to time. It provides both the speed and the direction of the particle's motion, which are essential for analyzing motion and subsequent interactions with forces.

Force

Force is an external vector quantity that can cause a particle to accelerate. In mechanics, forces are applied to objects to change their state of motion, and understanding their impact is key to applying Newton's laws of motion.

Dot Product

The dot product is an algebraic operation that multiplies corresponding components of two vectors and then sums the results, yielding a scalar. It is particularly useful in calculating work and power, where it combines force and displacement or velocity vectors.

Power

Power is defined as the rate at which work is done or energy is transferred in a system. In the context of mechanics, it is computed as the dot product of the force vector and the velocity vector, providing an instantaneous measure of how quickly energy is being used or converted.

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