Problem 29 - College Physics

Question

A \(0.45-\mathrm{kg}\) piece of putty is dropped from a height of \(2.5 \mathrm{~m}\) above a flat surface. When it hits the surface, the putty comes to rest in \(0.30 \mathrm{~s}\). What is the average force exerted on the putty by the surface?

Step-by-Step Solution

Answer

The average force exerted on the putty is \(-10.5 \, \text{N}\).

Step 1: Determine the Velocity at Impact

First, we calculate the velocity of the putty just before it strikes the surface. We use the formula for the final velocity of an object falling under gravity: \( v = \sqrt{2gh} \), where \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity and \( h = 2.5 \, \text{m} \) is the height. \[v = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 2.5 \, \text{m}} = \sqrt{49} = 7 \, \text{m/s}\]Thus, the velocity at impact is 7 m/s.

Step 2: Calculate the Change in Momentum

To find the change in momentum, we use the formula \( \Delta p = m \cdot \Delta v \), where \( m = 0.45 \, \text{kg} \) is the mass of the putty and \( \Delta v = -7 \, \text{m/s} \) (since it comes to rest, the final velocity is 0 and initial velocity is 7m/s).\[\Delta p = 0.45 \, \text{kg} \times (0 - 7 \, \text{m/s}) = -3.15 \, \text{kg} \cdot \text{m/s}\]The change in momentum is \(-3.15 \, \text{kg}\cdot\text{m/s}\).

Step 3: Calculate the Average Force

The average force exerted on the putty can be calculated from the change in momentum using the impulse-momentum theorem: \( \Delta p = F_{\text{avg}} \cdot \Delta t \). Solving for \( F_{\text{avg}} \), we use \( \Delta t = 0.30 \, \text{s} \).\[F_{\text{avg}} = \frac{\Delta p}{\Delta t} = \frac{-3.15 \, \text{kg} \cdot \text{m/s}}{0.30 \, \text{s}} = -10.5 \, \text{N}\]The average force exerted on the putty by the surface is \(-10.5 \, \text{N}\). The negative sign indicates direction opposite to the putty's motion.

Key Concepts

Impulse

Impulse is a fundamental concept in physics that explains the change in momentum of an object when a force is applied over a period of time. In simple terms, impulse equals the force multiplied by the time the force acts on an object. The formula for impulse is given by \[ \text{Impulse} = F \times \Delta t \].

• This concept is crucial in understanding how forces affect motion.
• Impulse provides insight into how much a force has changed the object's velocity.

In the given exercise, the impulse is the product of the average force exerted by the surface and the time duration of impact, which ultimately results in the putty coming to rest. Calculating impulse helps us determine the force characteristics needed to change the putty's momentum from its initial moving state to a state of rest.

Force

Force is a vector quantity described by both magnitude and direction. It is the influence that causes an object to undergo a particular motion or change shape. The formula for calculating force based on Newton's second law of motion is \[ F = m \times a \], where \( m \) is the mass and \( a \) is acceleration.

• In the world around us, force is observable in actions like pushing, pulling, or lifting.
• For moving objects, forces not only alter their speed but also their direction.

In the context of the given exercise, force is what brings the piece of putty to a stop when it hits the surface. Here, the average force exerted on the putty by the ground is what changes its momentum over the impact time, emphasizing the significance of understanding force application.

Kinematics

Kinematics is the study of motion without considering the forces that cause it. It involves analyzing positions, velocities, and accelerations of objects. This area of physics allows us to describe motion using mathematical formulas, such as \[ v = u + at \] for velocity and \[ s = ut + \frac{1}{2}at^2 \] for displacement, where \( u \) is the initial velocity, \( v \) is the final velocity, \( a \) is acceleration, \( t \) is time, and \( s \) is displacement.

• Kinematics focuses on the geometry of motion and is foundational for precise predictions in physics.

In the problem, kinematics is used to calculate the velocity of the putty just before impact, employing the formula \[ v = \sqrt{2gh} \]. Accurate velocity calculation is essential as it ties directly into momentum and impulse considerations.

Acceleration due to gravity

Acceleration due to gravity, represented by \( g \), is a constant that denotes the acceleration of an object caused by Earth's gravitational pull. Its standard value is approximately \[ g = 9.8 \, \text{m/s}^2 \].

• This acceleration affects all objects equally, regardless of their mass.
• It is responsible for the parabolic trajectory of projectiles and free-falling objects.

In the exercise, the acceleration due to gravity is what causes the putty to accelerate as it falls towards the ground. By using \( g \) in calculations such as \[ v = \sqrt{2gh} \], we determine the velocity of the putty at the moment just before it impacts the surface. Understanding \( g \) is crucial for predicting the behavior of falling objects.