Question

A force acts on a \(2.80-\mathrm{kg}\) particle in such a way that the position of the particle as a function of time is given by \(x=\) \((3.0 \mathrm{~m} / \mathrm{s}) t-\left(4.0 \mathrm{~m} / \mathrm{s}^{2}\right) t^{2}+\left(1.0 \mathrm{~m} / \mathrm{s}^{3}\right) t^{3} .(a)\) Find the work done by the force during the first \(4.0 \mathrm{~s}\). (b) At what instantaneous rate is the force doing work on the particle at the in\(\operatorname{stan} t=3.0 \mathrm{~s} ?\)

Short answer

The work done by the force on the particle during the first 4.0 s is calculated from the change in kinetic energy of the particle. The force doing work on the particle at an instantaneous rate at t = 3.0 s is determined by calculating the dot product of the force vector and the velocity vector.

Step-by-step solution

Find the velocity of the particle

To find the velocity of the particle, differentiate the position function, \(x=(3.0 \mathrm{~m}/\mathrm{s}) t - (4.0 \mathrm{~m}/\mathrm{s}^{2}) t^{2} + (1.0\mathrm{~m}/\mathrm{s}^{3}) t^{3}\), with respect to time. The resulting equation, \(v(t)= dx/dt =(3.0 \mathrm{~m}) -(8.0 \mathrm{~m}/\mathrm{s}) t + (3.0 \mathrm{~m}/\mathrm{s}^{2}) t^{2}\), will give us the instantaneous velocity of the particle at any given point in time.

Calculate the kinetic energy

The kinetic energy (\(KE\)) of the particle at any time `t` can be given as \(KE=1/2 m v^{2}\). Calculate the kinetic energy of the particle at the beginning (t = 0 s) and at the end of 4.0 s using the equation for velocity obtained in the previous step. The difference in kinetic energy will give us the work done on the particle, as \(Work = KE_{final} - KE_{initial}\).

Find the acceleration of the particle

To find the acceleration of the particle, further differentiate the velocity function with respect to time. \(a(t)= dv/dt =-(8.0 \mathrm{~m}/\mathrm{s}) + (6.0 \mathrm{~m}/\mathrm{s}^{2}) t\). This would give us the instantaneous acceleration of the particle at any given point in time.

Calculate the instantaneous force

The force exerted on the particle at any time `t` can be given by \(F=ma\), where `m` is the mass of the particle and `a` is the acceleration. Calculate the instantaneous force at t = 3.0 s using the equation for acceleration obtained in the previous step.

Calculate the instantaneous rate of work

Finally, the rate at which the force is doing work on a particle is given by the power, which is the dot product of the force vector and the velocity vector. Calculate the instantaneous power at t = 3.0 s using the equation for force and velocity obtained in previous steps.

Key concepts

Kinetic Energy

Kinetic energy (KE) is a form of energy associated with the motion of a particle or object. It is quantified by the equation \( KE = \frac{1}{2} m v^2 \), where \( m \) represents the mass of the object and \( v \) stands for its velocity. The kinetic energy of an object increases with its speed; the faster the object moves, the more kinetic energy it possesses.

Understanding kinetic energy is crucial when discussing work done by a force. In physics, work (W) is defined as the energy transferred to or from an object via the application of a force along a displacement. The relationship between work and kinetic energy is known as the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy: \( W = KE_{final} - KE_{initial} \).

For the exercise, by calculating the kinetic energy at two distinct times and taking their difference, we can determine the total work done on the particle within that time interval. This concept directly relates force to the change in motion, providing a bridge between kinematics and dynamics.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. Unlike average velocity, which describes how fast an object is moving over a period of time, instantaneous velocity gives us the speed and direction of an object at any singular point in time.

To find the instantaneous velocity, we differentiate the position function with respect to time, a process which gives us the velocity function. Mathematically, \( v(t) = \frac{dx}{dt} \). In our exercise, the position function of the particle is a polynomial, which when differentiated yields a straightforward expression for instantaneous velocity at any time \( t \).

Knowing the instantaneous velocity is essential for calculating the instantaneous power and kinetic energy, which in turn helps us to understand the dynamic behavior of objects under the influence of varying forces.

Instantaneous Acceleration

Instantaneous acceleration is the rate at which an object's velocity changes at a specific instant in time. In simpler terms, it tells us how quickly the velocity of an object is increasing or decreasing at a particular moment.

To find this acceleration, we differentiate the velocity function with respect to time: \( a(t) = \frac{dv}{dt} \). This second derivative of the position function with respect to time provides us with the instantaneous acceleration at any given point in time. In the context of the exercise, the given particle’s acceleration function can be obtained after the velocity function is differentiated.

The instantaneous acceleration is directly related to the instantaneous force exerted on an object through Newton's second law of motion, which states that \( F = ma \), where \( F \) is the force and \( a \) is the acceleration. This law is fundamental to understanding how forces cause changes in motion.

Instantaneous Force

Instantaneous force refers to the force acting on an object at a particular instant. Newton's second law (\( F = ma \) ) connects force directly with mass \( m \) and acceleration \( a \) at any given time. To calculate the instantaneous force from our provided exercise, we use the mass of the particle and its instantaneous acceleration at a specific time.

The concept of instantaneous force allows us to infer the immediate impact that a force has on an object's motion. In real-world applications, knowing the instantaneous force can be pivotal for engineering tasks, where precise control and timing are necessary.

Instantaneous Power

Instantaneous power is the rate at which work is done at a specific moment in time. To calculate power, we can use the equation \( P = F \cdot v \), where \( P \) is power, \( F \) is force, and \( v \) is velocity. In the context of the exercise, the instantaneous power is found by taking the dot product of the force vector and the velocity vector at a given moment, which in this case is at \( t = 3.0 \) seconds.

The concept of instantaneous power is essential in various fields, including mechanical engineering and electronics, as it allows for understanding the energy transfer rate from forces acting on objects and systems. Instantaneous power is also a critical factor in optimizing the performance of machines and devices by knowing how much power they can deliver or absorb at any given instant.