Problem 48 - University Physics with Modern Physics

Question

A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of \(a=2.80 \mathrm{m} / \mathrm{s}^{2}\) . A worker assists the cart by pushing on the crate with a force that is eastward and has magnitude that depends on time according to \(F(t)=\) \((5.40 \mathrm{N} / \mathrm{s}) t .\) What is the instantaneous power supplied by this force at \(t=5.00 \mathrm{s} ?\)

Step-by-Step Solution

Answer

The instantaneous power at \(t=5.00\) s is 378 W.

Step 1: Understand the Problem

We need to find the instantaneous power supplied by a force at a specific time. The force and the motion are both eastward, and the force magnitude depends on time.

Step 2: Identify Given Variables and Formulae

We know the force applied is given by \(F(t) = (5.40 \text{ N/s})t\). Power is defined as the product of force and velocity, \(P = F \cdot v\). We will need to find the velocity to compute power.

Step 3: Calculate Velocity at t=5.00 s

Since the cart starts from rest, we can use the kinematic equation for velocity under constant acceleration: \(v = a \cdot t\). Given \(a = 2.80 \text{ m/s}^2\), we substitute the values to find \(v = 2.80 \times 5.00 = 14.00 \text{ m/s}\).

Step 4: Determine Force at t=5.00 s

The force at \(t=5.00\) seconds can be found by substituting the time into the force equation: \(F(5.00) = 5.40 \times 5.00 = 27.00 \text{ N}\).

Step 5: Calculate Instantaneous Power at t=5.00 s

With the force and velocity known, we use the power formula: \(P = F \cdot v\). Substitute \(F=27.00\text{ N}\) and \(v=14.00\text{ m/s}\) to get \(P = 27.00 \times 14.00 = 378.00 \text{ W}\).

Key Concepts

Force and Velocity

Force and velocity are key concepts in understanding how objects move in physics. Force is any interaction that, when unopposed, will change the motion of an object. In the scenario, the worker applies a force that depends on time, described by the equation \( F(t) = (5.40 \, \text{N/s})t \). This indicates that as time increases, the force on the crate increases linearly.
Velocity, on the other hand, describes the speed of the crate in a specific direction. In this case, it's eastward as the cart and crate move. The velocity is related to both force and acceleration as per Newton's Second Law, which states that an object will only accelerate if there is a resultant force acting on it.

• Force affects how quickly an object moves in a given direction.
• Velocity is the outcome of integrating force over time under a constant acceleration condition.

Combining force and velocity helps in calculating instantaneous power, which is important in determining how much work is done by the force at a specific moment.

Motion with Constant Acceleration

In the exercise, the crate moves with a constant acceleration of \(2.80 \, \text{m/s}^2\). Motion with constant acceleration is a fundamental concept where an object's velocity changes at a steady rate.
This type of motion can be easily analyzed using kinematic equations, which allow us to find variables like velocity, displacement, or time. In situations of constant acceleration, you assume the acceleration does not change with time, simplifying the calculations.

• An object starting from rest with constant acceleration will have its velocity increase linearly with time.
• The acceleration remains constant, meaning its magnitude or direction doesn't change, leading to predictable motion.

This predictability allows us to use the formula \( v = a \cdot t \) to calculate velocity at any given time \( t \), which is useful in determining other dynamics like instantaneous power.

Kinematic Equations

Kinematic equations are a set of equations that describe the motion of an object under the influence of constant acceleration. They relate variables such as displacement, initial velocity, final velocity, acceleration, and time. The main kinematic equations are:

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)

In the context of our exercise, we specifically used the equation \( v = a \cdot t \) to find the velocity of the cart at 5.00 seconds. Here, \( u \) (initial velocity) is zero because the cart starts from rest.
The kinematic equations make solving for unknown variables straightforward when motion parameters like acceleration are constant, as they incorporate both time and acceleration succinctly. As a result, you can easily make predictions about future positions or velocities of moving objects.

Newtons Second Law

Newton's Second Law is a fundamental principle in physics that explains how the velocity of an object changes when it is subject to an external force. The law can be expressed as \( F = ma \), where \( F \) is the net force applied to an object, \( m \) is its mass, and \( a \) is its acceleration.
In this exercise, the law is implicitly used as it supports finding the velocity and force applied to the crate over time.

• The worker's force on the crate changes its motion by increasing its velocity eastward.
• This law helps in understanding why an object accelerates when a force is applied and doesn't change if the applied force is absent or balanced by another force.

Through the application of Newton's Second Law, we can predict how the changes in force over time (in this case, force increasing with time) affect the motion of objects, such as the cart and crate in the problem.