work and energy Questions


Between time $t=0\, {\rm s}$ and $t=8,{\rm s}$, a force ${{\vec{F}=\left(3\hat{i}-4.5\hat{j}\right)\,{\rm N}}}$ moves a $8.3\, {\rm kg}$ object along a trajectory $\Delta \vec{r}=(2.5\hat{i}-2\hat{j}){\rm m}$. How much work is done by this force?


The work done by a constant force $\vec{F}$ over a displacement $\vec{r}$ is $W=\vec{F}.\vec{r}=F\left|r\right|{\cos  \theta\ }$ where $\theta$ is the angle between the force and displacement.  So 
\[W=\vec{F}.\Delta \vec{r}=F_x\Delta x+F_y\Delta y=\left(3\right)\left(2.5\right)+\left(-4.5\right)\left(-2\right)=16.5\, {\rm J}\] 

A roller-coaster car moves around a vertical circular loop of radius$R=10.0\,{\rm m}$.
(a) What speed must the car have so that it will just make it over the top without any assistant the track? 

(b) What speed will the car subsequently have at the bottom of the loop?

(c) What will be the normal force on a passenger at the bottom of the loop?

The drag force $F_D$ on the dragster is plotted as a function of distance s below. What is the magnitude of the work done by the drag force after the dragster has traveled $400\, {\rm m}$? 

A $200\, {\rm g}$ rubber ball is tied to a $1.0\, {\rm m}$ long string and released from rest at angle $\theta$. It swings down and at the very bottom has a perfectly elastic collision with a $1.0\,{\rm kg}$ block. The block is resting on a frictionless surface and is connected to a $20\,{\rm cm}$ long spring of spring constant $2000\, {\rm N/m}$. After the collision, the spring compresses a maximum distance of $2.0\,{\rm cm}$. from what angle was the rubber ball released?

A Skier, whose mass is $70\, {\rm kg}$, stands at the top of a $10{}^\circ $ slope on her new frictionless skis. A strong horizontal wind blows against him with a force of $50\, {\rm N}$. Without using Newton's laws of motion, find skier's speed after traveling $100\, {\rm m}$ down the slope.

A bullet of mass $m$ and speed $v$ passes completely through a pendulum bob of mass $M$. The bullet emerges with speed of $v/2$. The pendulum bob is suspended by a stiff rod of length $l$ and negligible mass. What is the minimum value of $v$ such that the pendulum bob will barely swing through a complete vertical circle?

A block of mass $M$ attached to a horizontal spring with force constant $k$ is moving in SHM with amplitude $A_1$. As the block passes through its equilibrium position, a lump of putty of mass $m$ is dropped from a small height and sticks to it. 

(a) Find the new amplitude and period of the motion.

(b) Repeat first part, find the new amplitude and period of the motion, if the putty is dropped from a small height onto the block and sticks to it when it is at one end of its path.

Tarzan is in the path of a pack of stampeding elephants when Jane swings in to the rescue on a rope vine, hauling him off to safety. The length of the vine is $25\,{\rm m}$, and Jane starts her swings with the rope horizontal. If Jane's mass is $54\,{\rm kg}$, and Tarzan's mass is $82\,{\rm kg}$, to what height above the ground will the pair swing after she rescues him?

Released from rest at the same height, a thin spherical shell and solid sphere of the same mass $m$ and radius $R$ roll without slipping down an incline through the same vertical drop $H$. Each is moving horizontally as it leaves the ramp. The spherical shell hits the ground a horizontal distance $L$ from the end of the ramp and the solid sphere hits the ground a distance $l^{'}$ from the end of the ramp. Find the ratio $L^{'}/L$.

The magnitude of a single force acting on a particle of mass $m$ is given by $F=bx^2$, where $b$ is a constant. The particle starts from rest. After it travels a distance $L$, determines its 

(a) Kinetic energy and

(b) Speed

A box of mass $M$ is at rest at the bottom of a frictionless inclined plane. The box is attached to a string that pulls with a constant tension $T$. 

(a) Find the work done by the tension $T$ as the box moves through a distance $x$ along the plane.

(b) Find the speed of the box as a function of $x$.

(c) Determine the power delivered by the tension in the string as a function of $x$.


An Atwood's machine consists of masses $m_1$ and $m_2$, and a pulley of negligible mass and friction. Starting from rest, the speed of the two masses is $4.0\ {\rm m/s}$ at the end of $3.0\ {\rm s}$. At that time, the kinetic energy of the system is $80\ {\rm J}$ and each masses has moved a distance of $6.0\ {\rm m}$. Determine the values of $m_1$ and $m_2$.

A block of mass $m$ rests on an inclined plane. The coefficient of static friction between the block and the plane is $\mu_{s}$. A gradually increasing force is pulling down on the spring (force constant $k$). Find the potential energy $U$ of the spring at the moment the block begins to move.

A box of mass $m$ on the floor is connected to a horizontal spring of force constant $k$. The coefficient of kinetic friction between the box and the floor is $\mu_k$. The other end of the spring is connected to a wall. The spring is initially unstressed. If the box is pulled away from the wall a distance $d_0$ and released, the box slides toward the wall. Assume the box does not slide so far that the coils of the spring touch. 

(a) Obtain an expression for the distance $d_1$ the box slides before it first comes to a stop. 

(b) Assuming $d_1>d_0$, obtain an expression for the speed of the box when it has slid a distance $d_0$ following the release.

(c) Obtain the special value of $\mu_k$ such that $d_1=d_0$.

A $5.00\,{\rm kg}$ block is firmly attached to a $120\,{\rm N/m}$ spring. The block is initially at rest and the entire setup is on a frictionless surface. A rope inclined at $36.9{}^\circ $ above the horizontal is used to slowly pull the block until the block-spring-rope system is again at rest. At this point, the spring is stretched by $40.0\, {\rm cm}$.

(a) Calculate the tension in the rope at this point( spring stretched by $40.0\, {\rm cm}$)

(b) Calculate the work done by the tension in the rope as the block moves from its initial position to this point.( spring stretched by $40.0\, {\rm cm}$)

(c) Suddenly the rope breaks. Calculate the speed of the block at the point where the spring is once again upstretched-uncompressed.

In the system drawn below, the coefficient of kinetic friction between the $6.00\, {\rm kg}$ block and the horizontal surface is $0.15$, and the entire system is being held at rest (by someone who is holding the hanging $2.00\, {\rm kg}$ block in place). This person releases the block. Calculate the speed of each block after the $2.00\, {\rm kg}$ block has fallen $30.0\, {\rm cm}$.

An $80\, {\rm g}$ arrow is fired from a bow whose string exerts an average force of $95\, {\rm N}$ on the arrow over a distance of $80\, {\rm cm}$. What is the speed of the arrow as it leaves the bow?

A force $\vec{F}=12\ \hat{i}-10\hat{j}\ {\rm (N)}$ acts on an object. How much work does this force do as the objects moves from the origin to the point $\vec{r}=12\hat{i}+11\hat{j}\ {\rm (m)}$

In the figure, a constant external force $P=160\, {\rm N}$ is applied to a $20\, {\rm kg}$ box, which is on a rough horizontal surface. While the force pushes the box a distance of $8\, {\rm m}$, the speed changes from $0.5\, {\rm m/s}$ to $2.6\, {\rm m/s}$. What is the work done by friction during this process?

A net force along the $x$ axis $F\left(x\right)=-C+Dx^2$ is applied to a mass of $m$ that is initially at the origin moving in the $-x$ direction with a speed of $v_0$. What is the speed of the object when it reaches a point $x_f$?

A body of mass $M$ is seated on top of a hemispherical mound of ice of radius $R$ as shown below. He starts to slide down the ice and eventually flies off the mound of ice. The ice is frictionless. 

(a) Draw a free body diagram for the boy when he is at point $P$.

(b) At angle $\theta$, what is the boy's velocity?

(c) What is $\theta_0$ the angle at which the boy flies off the ice mound?

A uniform solid sphere is rolling without slipping along a horizontal surface with a speed of $4.5\, {\rm m/s}$ when it starts up a ramp that makes an angle of $25{}^\circ $ with the horizontal. What is the speed of the sphere after it has rolled $3.00\, {\rm m}$ up the ramp, measured along the surface of the ramp?

A huge cannon is assembled on an airless planet having insignificant axial spin. The planet has a radius of $5\times {10}^6\, {\rm m}$ and a mass of $3.95\times {10}^{23}\,{\rm \ kg}$. The cannon fires a projectile straight up at $2000\, {\rm m/s}$. An observation satellite orbits the planet at a height of $1000\, {\rm km}$. What is the projectile's speed as it passes the satellite?

A roller coaster cart rolls from rest down a ${\rm 50.0\ m}$ tall hill and then goes around a circular vertical loop-the-loop of radius ${\rm 15.0\ m}$ as shown at right.

(a) How fast is the cart going when it gets to the top of the loop?

(b) If the mass of the cart (including all of its passengers) is ${\rm 1.20\times }{{\rm 10}}^{{\rm 3}}{\rm \ kg}$, what is the magnitude of the normal force that acts on the cart at the top of the loop? 

A uniform solid sphere of radius $r$starts from rest at a height $h$ and rolls without slipping along the loop the loop track of radius $R$ as shown in the figure. ($r\ll R$).

(a) Draw the free body diagram for the sphere when it is at the top of the loop (point A) and moving fast enough to stay on the truck.

(b) What is the smallest height $h$ for which the sphere will not leave the truck at the top?


An $8\, {\rm kg}$ block is released from rest, $v_1=0\, {\rm m/s}$, on a rough incline. The block moves a distance of $1.6\, {\rm m}$ down the incline, in a time interval of $0.8\, {\rm s}$, and acquires a velocity of $v_2=4.0\, {\rm m/s}$. 

(a) Calculate the works done by the weight, friction and normal forces.

(b) What is the average rate at which the block gains kinetic energy during the $0.8\, {\rm s}$ time interval?

Imagine a toy gun in which a ball is shot out when a spring is released. The force constant of the spring is $10\, {\rm N/m}$, and it is compressed by $0.05\, {\rm m}$. The mass of the ball is $0.02\, {\rm kg}$. If no energy is lost to friction, approximately what is the speed of the ball when it is shot out?

A $25\ {\rm kg}$ child plays on a swing having support ropes that are $3.0\ {\rm m}$ long. A friend pulls her back until the ropes are $45{}^\circ $ from the vertical and releases her from rest.

(a) What is the height of the child above her lowest point, at the moment she is released?

(b) What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing?

(c) How fast will she be moving at the bottom of the swing?

A uniform solid cylinder starts from rest at a height of $1\ {\rm m}$ and rolls without slipping down a plane inclined at an angle of $20{}^\circ $ from the horizontal as shown.

(a) What is the speed of its center of mass when it reaches the bottom of the incline?

(b) What is the magnitude of the acceleration of its center of mass?

(c) What minimum coefficient of friction is required to prevent the cylinder from slipping?

Two rocks are thrown from a building $11\,{\rm m}$ high, each with a speed of $5\ {\rm m/s}$. one is thrown vertically upwards, the other horizontally. What is the speed at which each rock will hit the ground?

work and energy
Category : work and energy

Most useful formula in Work-Energy:

Definition of Work: $W=\vec F \cdot \vec s=Fs \,\rm{cos} \phi$

Kinetic energy: $K=\frac{1}{2}mv^2$

Work-Energy theorem : $W_{tot}=\Delta K=K_2-K_1$

Power: $P_{av}=\frac{\Delta W}{\Delta t}=\vec F \cdot \vec v$

Number Of Questions : 30