Question

While an elevator of mass \(2530 \mathrm{kg}\) moves upward, the force exerted by the cable is \(33.6 \mathrm{kN} .\) (a) What is the acceleration of the elevator? (b) If at some point in the motion the velocity of the elevator is \(1.20 \mathrm{m} / \mathrm{s}\) upward, what is the elevator's velocity 4.00 s later?

Short answer

(a) Acceleration is approximately 3.48 m/s² upward. (b) Velocity after 4.00 s is approximately 15.12 m/s upward.

Step-by-step solution

Understanding Forces and Newton's Second Law

The problem states that the force exerted by the cable on the elevator is known, as well as the mass of the elevator. We know from Newton's Second Law that \[ F = ma \] where \( F \) is the total force, \( m \) is the mass, and \( a \) is the acceleration of the elevator. Importantly, the total force is the net force acting on the system (elevator), which in this case includes the upward force by the cable and the downward force due to gravity.

Calculate Gravitational Force

Calculate the force of gravity acting on the elevator:\[ F_{gravity} = mg \]where \( g = 9.8 \, \mathrm{m/s^2} \) (acceleration due to gravity) and \( m = 2530 \, \mathrm{kg} \).Substitute the values in:\[ F_{gravity} = 2530 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 24794 \, \mathrm{N} \].

Determine the Net Force

The net force \( F_{net} \) on the elevator is the difference between the upward force exerted by the cable and the gravitational force:\[ F_{net} = F_{cable} - F_{gravity} \]where \( F_{cable} = 33600 \, \mathrm{N} \).Therefore,\[ F_{net} = 33600 \, \mathrm{N} - 24794 \, \mathrm{N} = 8806 \, \mathrm{N}. \]

Calculate Acceleration of the Elevator

Using Newton's Second Law,\[ a = \frac{F_{net}}{m} \]Substitute the values:\[ a = \frac{8806 \, \mathrm{N}}{2530 \, \mathrm{kg}} \approx 3.48 \, \mathrm{m/s^2}. \]This is the acceleration of the elevator upwards.

Understand the Motion for Velocity Calculation

Now that we have the acceleration, we can calculate the change in velocity over a given time period. We already know that the initial velocity \( v_0 \) of the elevator is \( 1.20 \, \mathrm{m/s} \) upwards, and we want to find the velocity \( v \) after \( 4.00 \, \mathrm{s} \).

Use the Kinematic Equation to Find Final Velocity

The kinematic equation to find the final velocity is:\[ v = v_0 + at \]where \( v_0 = 1.20 \, \mathrm{m/s} \), \( a = 3.48 \, \mathrm{m/s^2} \), and \( t = 4.00 \, \mathrm{s} \).Substitute the values:\[ v = 1.20 \, \mathrm{m/s} + (3.48 \, \mathrm{m/s^2} \times 4.00 \, \mathrm{s}) \]\[ v = 1.20 \, \mathrm{m/s} + 13.92 \, \mathrm{m/s} \approx 15.12 \, \mathrm{m/s}. \]The elevator's velocity after 4.00 seconds is approximately \( 15.12 \, \mathrm{m/s} \) upwards.

Key concepts

Understanding Acceleration

Acceleration is a crucial concept in physics, signifying the rate at which an object's velocity changes over time. It is directly influenced by the net force acting on the object and its mass, as described by Newton's Second Law. Acceleration can result in an object speeding up, slowing down, or changing direction. In the context of the elevator problem, we calculate acceleration to determine how quickly the elevator is moving upwards.
To find the acceleration, we use the formula:

• Newton's Second Law: \( F = ma \), which can be rearranged to \( a = \frac{F_{net}}{m} \).
• In our exercise, \( a \) (the acceleration) was calculated using the net force acting on the elevator and its mass.

This demonstrates how acceleration is vital for understanding motion, as it helps predict how quickly an object's velocity changes.

Net Force Explained

Net force is the total force acting on an object, considering all the individual forces applied. It's a determining factor for acceleration; without a net force, there's no change in an object's velocity. The net force considers both the magnitude and direction of all acting forces.
For the elevator scenario:

• The upward force exerted by the cable is counteracted by the gravitational force pulling the elevator down.
• We calculate the net force as the difference between these two forces: \( F_{net} = F_{cable} - F_{gravity} \).
• In simpler terms, it's like subtracting the weight of the elevator from the force pulling it up.

Understanding net force allows us to determine how much of a push or pull is effectively moving the object, facilitating a deeper grasp of the forces underlying motion.

Kinematic Equations for Motion

Kinematic equations are used to describe the motion of objects without necessarily considering the forces that cause this motion. They relate variables such as velocity, acceleration, time, and displacement to give a comprehensive picture of motion.
In solving the elevator problem:

• We applied the kinematic equation \( v = v_0 + at \) to find the final velocity after a certain time has passed.
• Given our initial velocity, acceleration, and time duration, we plugged these into the equation to find the elevator's velocity after 4 seconds.
• This demonstrates how kinematic equations can predict future motion based on current motion parameters and changes.

These equations are powerful tools in physics, offering insights into how objects move over time from initial to final states without delving into the complexities of force interactions.