Motion along a straight line | Practice Problems, Examples & Solutions

Displacement and Time

Engineering Mechanics: Statics and Dynamics

The uniform rod of mass $m$ is supported by a pin at $A$ and a spring at $B$. If $B$ is given a small sideward displacement and released, determine the natural period of vibration.

Vibrations

01:34

Engineering Mechanics: Statics and Dynamics

A $3-\mathrm{kg}$ block is suspended from a spring having a stiffness of $k=200 \mathrm{N} / \mathrm{m}$. If the block is pushed $50 \mathrm{mm}$ upward from its equilibrium position and then released from rest, determine the equation that describes the motion. What are the amplitude and the natural frequency of the vibration? Assume that positive displacement is downward.

Vibrations

02:32

Engineering Mechanics: Statics and Dynamics

A $6-$ lb weight is suspended from a spring having a stiffness $k=3$ lb/in. If the weight is given an upward velocity of $20 \mathrm{ft} / \mathrm{s}$ when it is 2 in. above its equilibrium position, determine the equation which describes the motion and the maximum upward displacement of the weight, measured from the equilibrium position. Assume positive displacement is downward.

Vibrations

Average Velocity

515 Practice Problems

02:51

Engineering Mechanics: Statics and Dynamics

A spring is stretched $175 \mathrm{mm}$ by an 8-kg block. If the block is displaced $100 \mathrm{mm}$ downward from its equilibrium position and given a downward velocity of $1.50 \mathrm{m} / \mathrm{s}$ determine the differential equation which describes the motion. Assume that positive displacement is downward. Also, determine the position of the block when $t=0.22 \mathrm{s}$.

Vibrations

01:23

Engineering Mechanics: Statics and Dynamics

The pulley has a weight of 8 lb and may be treated as a thin disk. A cord wrapped over its surface is subjected to forces $T_{A}=4$ lb and $T_{B}=5$ lb. Determine the angular velocity of the pulley when $t=4$ s if it starts from rest when $t=0 .$ Neglect the mass of the cord.

Planar Kinetics of a Rigid Body: Impulse and Momentum

07:07

Engineering Mechanics: Statics and Dynamics

The disk rotates about the shaft $S,$ while the shaft is turning about the $z$ axis at a rate of $\omega_{z}=4 \mathrm{rad} / \mathrm{s}$, which is increasing at 2 rad $/ \mathrm{s}^{2}$. Determine the velocity and acceleration of point $B$ on the disk at the instant shown. No slipping occurs.

Three-Dimensional Kinematics of a Rigid Body

Instantaneous Velocity

139 Practice Problems

01:44

Engineering Mechanics: Statics and Dynamics

At a given instant the body of mass $m$ has an angular velocity $\omega$ and its mass center has a velocity $\mathbf{v}_{G} .$ Show that its kinetic energy can be represented as $T=\frac{1}{2} I_{I C} \omega^{2}$, where $I_{I C}$ is the moment of inertia of the body determined about the instantaneous axis of yero velocity, located a distance $r_{G / K}$ from the mass center as shown.

Planar Kinetics of a Rigid Body: Work and Energy

02:43

Engineering Mechanics: Statics and Dynamics

When a particle falls through the air, its initial acceleration $a=g$ diminishes until it is zero, and thereafter it falls at a constant or terminal velocity $v_{f} .$ If this variation of the acceleration can be expressed as $a=\left(g / v^{2}_{f}\right)\left(v_{f}^{2}-v^{2}\right),$ determine the time needed for the velocity to become $v=v_{f} / 2 .$ Initially the particle falls from rest.

Kinematics of a Particle

04:57

Engineering Mechanics: Statics and Dynamics

If the effects of atmospheric resistance are accounted for, a freely falling body has an acceleration defined by the equation $a=9.81\left[1-v^{2}\left(10^{-4}\right)\right] \mathrm{m} / \mathrm{s}^{2}$
where $v$ is in $\mathrm{m} / \mathrm{s}$ and the positive direction is downward. If the body is released from rest at a very high altitude, determine (a) the velocity when $t=5 \mathrm{s}$, and (b) the body's terminal or maximum attainable velocity $(\text { as } t \rightarrow \infty)$

Kinematics of a Particle

Average and Instantaneous Acceleration

424 Practice Problems

18:46

Engineering Mechanics: Statics and Dynamics

The disk rotates about the $z$ axis at a constant rate $\omega_{z}=0.5 \mathrm{rad} / \mathrm{s}$ without slipping on the horizontal plane. Determine the velocity and the acceleration of point $A$ on the disk.

Three-Dimensional Kinematics of a Rigid Body

04:28

Engineering Mechanics: Statics and Dynamics

The acceleration of a particle along a straight line is defined by $a=(2 t-9) \mathrm{m} / \mathrm{s}^{2},$ where $t$ is in seconds. At $t=0, s=1 \mathrm{m}$ and $v=10 \mathrm{m} / \mathrm{s} .$ When $t=9 \mathrm{s},$ determine
(a) the particle's position,
(b) the total distance traveled, and (c) the velocity.

Kinematics of a Particle

01:21

Engineering Mechanics: Statics and Dynamics

The motor pulls on the cable at $A$ with a force $F=\left(30+t^{2}\right)$ Ib, where $t$ is in seconds. If the 34 -lb crate is originally on the ground at $t=0,$ determine its speed in $t=4 \mathrm{s}$ Neglect the mass of the cable and pulleys. Hint: First find the time needed to begin lifting the crate.

Kinetics of a Particle: Impulse and Momentum

Motion with Constant Acceleration

219 Practice Problems

01:51

Engineering Mechanics: Statics and Dynamics

A spring has a stiffness of $800 \mathrm{N} / \mathrm{m} .$ If a 2 -kg block is attached to the spring, pushed $50 \mathrm{mm}$ above its equilibrium position, and released from rest, determine the equation that describes the block's motion. Assume that positive displacement is downward.

Vibrations

03:32

Engineering Mechanics: Statics and Dynamics

The acceleration of a particle traveling along a straight line is $a=\frac{1}{4} s^{1 / 2} \mathrm{m} / \mathrm{s}^{2},$ where $s$ is in meters. If $v=0$
$s=1 \mathrm{m}$ when $t=0,$ determine the particle's velocity at $s=2 \mathrm{m}$

Kinematics of a Particle

03:31

Engineering Mechanics: Statics and Dynamics

A particle is moving along a straight line such that its acceleration is defined as $a=(-2 v) \mathrm{m} / \mathrm{s}^{2},$ where $v$ is in meters per second. If $v=20 \mathrm{m} / \mathrm{s}$ when $s=0$ and $t=0$ determine the particle's position, velocity, and acceleration as functions of time.

Kinematics of a Particle

Free Falling Bodies

111 Practice Problems

01:02

21st Century Astronomy

Astronauts in the International Space Station
a. have no mass.
b. have no energy.
c. are outside of Earth's gravitational field.
d. are in free fall.

Relativity and Black Holes

01:06

Principles of Physics a Calculus Based Text

A ball of mass $m$ falls from a height $h$ to the floor.
(a) Write the appropriate version of Equation 7.2 for the system of the ball and the Earth and use it to calculate the speed of the ball just before it strikes the Earth. (b) Write the appropriate version of Equation 7.2 for the system of the ball and use it to calculate the speed of the ball just before it strikes the Earth.

Conservation of Energy

05:31

University Physics Volume 1

A person drops a pebble of mass $m_{1}$ from a height $h,$ and it hits the floor with kinetic energy $K .$ The person drops another pebble of mass $m_{2}$ from a height of $2 h$, and it hits the floor with the same kinetic energy $K$. How do the masses of the pebbles compare?

Work and Kinetic Energy

Velocity and Position by Integration

36 Practice Problems

02:23

College Physics

On May $26,1934,$ a streamlined, stainless steel diesel train called the Zephyr set the world's nonstop long-distance speed record for trains. Its run from Denver to Chicago took 13 hours, 4 minutes, 58 seconds, and was witnessed by more than a million people along the route. The total distance traveled was $1633.8 \mathrm{km}$. What was its average speed in $\mathrm{km} / \mathrm{h}$ and m/s?

Kinematics

01:21

College Physics

Land west of the San Andreas fault in southern California is moving at an average velocity of about $6 \mathrm{cm} / \mathrm{y}$ northwest relative to land east of the fault. Los Angeles is west of the fault and may thus someday be at the same latitude as San Francisco, which is east of the fault. How far in the future will this occur if the displacement to be made is $590 \mathrm{km}$ northwest, assuming the motion remains constant?

Kinematics

01:13

College Physics

The North American and European continents are moving apart at a rate of about $3 \mathrm{cm} / \mathrm{y}$. At this rate how long will it take them to drift $500 \mathrm{km}$ farther apart than they are at present?

Kinematics