In the figure shown below, a block of mass  $m=5.0 \mathrm{kg}$ is at rest on a ramp. The coefficient of static friction between the block and the ramp is not known. Find the magnitude of the net force exerted by the ramp on the block.

In the figure shown below a small block of mass $m$ is sent sliding with velocity $v$ along a slab of mass$10 m$, starting at a distance of $l$ from the far end of the slab. The coefficient of kinetic friction between the slab and the floor is ${\mu }_1;$ that between the block and the slab is ${\mu }_2,$ with ${\mu }_2>11{\mu }_1$ . (a) Find the minimum value of $v$ such that the block reaches the far end of the slab. (b) For that value of $v$, how long does the block take to reach the far end?

In the figure shown below a force $\overrightarrow{P}$ acts on a block weighing $45 \mathrm{N}$. The block is initially at rest on a plane inclined at angle $\text{θ}=15°$ to the horizontal. The positive direction of the $x$-axis is up the plane. Between block and plane, the coefficient of static friction is ${\mu }_{\mathrm{s}}=0.50$ and the coefficient of kinetic friction is ${\mu }_{\mathrm{k}}=0.34$. In unit-vector notation, what is the frictional force on the block from the plane when $\overrightarrow{P}$ is (a) $(-5.0 \mathrm{N})\hat{\mathrm{i}}$ (b) $(-8.0 \mathrm{N})\hat{\mathrm{i}}$ and (c)$(-15 \mathrm{N})\hat{\mathrm{i}}$?

You testify as an expert witness in a case involving an accident in which car $A$ slid into the rear of car $B$, which was stopped at a red light along a road headed down a hill (shown in the figure below). You find that the slope of the hill is $\text{θ}=12.0°$. The cars were separated by distance $d=30.0 \mathrm{m}$ when the driver of car $A$ put the car into a slide (it lacked any automatic anti-lock braking system). The speed of car $A$ at the onset of braking was $v_0=18.0 \mathrm{m} {\mathrm{s}}^{-1}$. With what speed did car $A$ hit car $B$, if the coefficient of the kinetic friction was

(a) $0.60$ (dry road surface)?

(b) $0.10$ (road surface covered with wet leaves)?

 In figure, a box of Cheerios $\text{ (mass }{m}_{\mathrm{c}}=1.0\text{kg ) }$ and a box of Wheaties $\left(\text{ mass }{m}_{\mathrm{w}}=3.0\mathrm{kg}\right)$ are accelerated across a horizontal surface by a horizontal force $\overrightarrow{F}$ applied to the Cheerios box. The magnitude of the frictional force on the Cheerios box is $2.0 \mathrm{N}$, and the magnitude of the frictional force on the Wheaties box is $3.5 \mathrm{N}$. If the magnitude of $\overrightarrow{F}$ is$12 \mathrm{N}$, what is the magnitude of the force on the Wheaties box from the Cheerios box?

In a $15$ kg sled is attached to a $2.0$ kg  sand box by a string of negligible mass, wrapped over a pulley of negligible mass and friction. The coefficient of kinetic friction between the sled and table top is $0.040$ .Find (a) the acceleration of the sled and (b) the tension of the string.

In figure (a), a sled is held on an inclined plane by a cord pulling directly up the plane. The sled is to be on the verge of moving up the plane. In figure (b), the magnitude $F$ required of the cord's force on the sled is plotted versus a range of values for the coefficient of static friction ${\mu }_2$, between sled and plane: $F_1=2.0 \mathrm{N}, F_2=5.0 \mathrm{N},\text{ and }{\mu }_2=0.25$. At what angle $\theta$ is the plane inclined?

When the three blocks in are released from rest, they accelerate with a magnitude of$0.500 \mathrm{m}/{\mathrm{s}}^2$. Block $1$ has mass$M$, block $2$ has $2M$, and block $3$ has$2M$. What is the coefficient of kinetic friction between block $2$ and the table?

A $4.10 \mathrm{kg}$ block is pushed along a floor by a constant applied force that is horizontal and has a magnitude of $50 \mathrm{N}$. Figure gives the block's speed $v$ versus time $t$ as the block moves along an $x$-axis on the floor. The scale of the figure's vertical axis is set by $v_{\mathrm{s}}\mathit{=}5.0 \mathrm{m} {\mathrm{s}}^{-1}$. What is the coefficient of kinetic friction between the block and the floor?