The crate has a mass of $80 \mathrm{~kg}$ and is being towed by a chain which is always directed at $20^{\circ}$ from the horizontal. If the magnitude of $\mathbf{P}$ is increased until the crate begins to slide, determine the crate's initial acceleration if the coefficient of static friction is $\mu_s=0.5$ and the coefficient of kinetic friction is $\mu_k=0.3$.

Each of the three barges has a mass of $30 \mathrm{Mg}$, whereas the tugboat has a mass of $12 \mathrm{Mg}$. As the barges are being pulled forward with a constant velocity of $4 \mathrm{~m} / \mathrm{s}$, the tugboat must overcome the frictional resistance of the water, which is $2 \mathrm{kN}$ for each barge and $1.5 \mathrm{kN}$ for the tugboat. If the cable between $A$ and $B$ breaks, determine the acceleration of the tugboat.

The 10-lb block has an initial velocity of $10 \mathrm{ft} / \mathrm{s}$ on the smooth plane. If a force $F=(2.5 t) \mathrm{lb}$, where $t$ is in seconds, acts on the block for $3 \mathrm{~s}$, determine the final velocity of the block and the distance the block travels during this time.

The $75-\mathrm{kg}$ man climbs up the rope with an acceleration of $0.25 \mathrm{~m} / \mathrm{s}^2$, measured relative to the rope. Determine the tension in the rope and the acceleration of the $80-\mathrm{kg}$ block.

A small, frictionless car coasts down an inclined track with a $R$-radius circular loopthe-loop at its lower end. To traverse the loop without leaving the track, the car must begin at what height $H$ above the top of the loop?

Shown below are two carts connected by a cord that passes over a small frictionless pulley. Each cart rolls freely with negligible friction. Calculate the acceleration of the carts and the tension in the cord.

The pilot of the airplane executes a vertical loop which in part follows the path of a cardioid, $r = 200(1 + \cos\theta)$ m, where $\theta$ is in radians. If his speed at A is a constant $v_p$= 85 m/s, determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at A. He has a mass of 80 kg.

The force of the motor M on the cable is shown in the graph. Determine the velocity of the 400-kg crate A when t=2s.

The $400-\mathrm{kg}$ mine car is hoisted up the incline using the cable and motor $M$. For a short time, the force in the cable is $F=\left(3200 t^2\right) \mathrm{N}$, where $t$ is in seconds. If the car has an initial velocity $v_1=2 \mathrm{~m} / \mathrm{s}$ when $t=0$, determine the distance it moves up the plane when $t=2 \mathrm{~s}$.

Car B turns such that its speed is increased by $\left(a_t\right)_B=\left(0.5 e^t\right) m / s ^2$, where t is in seconds. If the car starts from rest when $\theta=0^{\circ}$, determine the magnitudes of its velocity and acceleration when the arm A B rotates $\theta=30^{\circ}$. Neglect the size of the car.

Car B turns such that its speed is increased by $\left(a_t\right)_B=\left(0.5 e^t\right) \mathrm{m} / \mathrm{s}^2$, where t is in seconds. If the car starts from rest when $\theta=0^{\circ}$, determine the magnitudes of its velocity and acceleration when t=2 s. Neglect the size of the car.

Car B turns such that its speed is increased by $\left(a_t\right)_B=\left(0.5 e^t\right) m / s ^2$, where t is in seconds. If the car starts from rest when $\theta=0^{\circ}$, determine the magnitudes of its velocity and acceleration when t=2 s. Neglect the size of the car.

Team $A$ and team $B$ consist of four members of total mass $320 \mathrm{~kg}$ and $330 \mathrm{~kg}$, respectively. If the average coefficients of static and kinetic friction for team $A$ are $\mu_n=0.52$ and $\mu_k=$ $0.5$, and for team $B, \mu_s=0.48$ and $\mu_k=0.46$, which team will win the tug of war? Also, what is the acceleration of the losing team if the winning team develops its maximum static friction force?

(a) Determine the critical nucleus size $r^*$ for homogeneous nucleation for precipitation of phase $\beta$ in a matrix of phase $\alpha$.Hint: The critical nucleus size occurs at the maximum in the expression for $\Delta G(r)$ in Equation 12-1. (b) Plot the total free energy change $\Delta G$ as a function of the radius of the precipitate. (c) Comment on the value of $r^*$ for homogeneous nucleation for solid-state precipitation when compared to the liquid to solid transformation.