Resistance of air acting on a falling body can be taken into account by the approximate relation for the acceleration:

$$a=\frac{d v}{d t}=g-k v,$$

where $k$ is a constant. Derive a formula for the velocity of the body as a function of time assuming it starts from rest $(v=0$ at $t=0)$. [Hint: Change variables by setting $u=g-k v$.]

When April provides an input force of 10 N, the drum key provides an output force of 15 N.

Is the output force acting in the same direction as the input force?

Calculate the mechanical advantage of the drum key.

Since the drum key increases April's input force, is her input distance greater or smaller than the key's output distance?

A long piece of wire with a mass of 0.100 kg and a total length of 4.00 m is used to make a square coil with a side of 0.100 m. The coil is hinged along a horizontal side, carries a 3.40-A current, and is placed in a vertical magnetic field with a magnitude of 0.010 0 T. Find the torque acting on the coil due to the magnetic force at equilibrium.

Two soccer players kick two identical balls at the same time. Player 1 sends his ball at an impressive 90 km/hr at an angle of $30 ^ { \circ }$ above the horizontal. Player 2 kicks at 70 km/hr at $45 ^ { \circ }$. Once in midair, which ball has a larger force acting on it

A half-full recycling bin has mass 3.0 kg and is pushed up a 40.0° incline with constant speed under the action of a 26-N force acting up and parallel to the incline. The incline has friction. What magnitude force must act up and parallel to the incline for the bin to move down the incline at constant velocity?

A long, straight wire lies on a horizontal table in the xy-plane and carries a current of 1.20 $\mu \mathrm { A }$ in the positive x-direction along the x-axis. A proton is traveling in the negative x-direction at speed $2.30 \times 10 ^ { 4 }$ m/s a distance d above the wire (i.e., z = d). What is the direction of the magnetic force acting on the proton?

The equation of the deflection curve for a simply supported beam is

$$v(x)=\frac{q_0}{24 E I}\left(2 L x^3-x^4-L^3 x\right)$$

(a) Derive the slope equation of the beam. (b) Derive the bending-moment equation of the beam. (c) Derive the shear-force equation of the beam. (d) Describe the loading acting on the beam.

At the instant the displacement of a 1.50 kg object relative to the origin is

$$\vec { d } = ( 2.00 \mathrm { m } ) \hat { \mathrm { i } } + ( 4.00 \mathrm { m } ) \hat { \mathrm { j } } - ( 3.00 \mathrm { m } ) \hat { \mathrm { k } }$$

, its velocity is

$$\vec { v } = - ( 6.00 \mathrm { m } / \mathrm { s } ) \hat { \mathrm { i } } + ( 3.00 \mathrm { m } / \mathrm { s } ) \hat { \mathrm { j } } + ( 3.00 \mathrm { m } / \mathrm { s } ) \hat { \mathrm { k } }$$

and it is subject to a force

$$\vec { F } = ( 6.00 \mathrm { N } ) \hat { \mathrm { i } } - ( 8.00 \mathrm { N } ) \hat { \mathrm { j } } + ( 4.00 \mathrm { N } ) \hat { \mathrm { k } }$$

. Find (a) the acceleration of the object, (b) the angular momentum of the object about the origin, (c) the torque about the origin acting on the object, and (d) the angle between the velocity of the object and the force acting on the object.

A rubber ball bounces. We'd like to understand how the bounces. a. A rubber ball has been dropped and is bouncing off the floor. Draw a motion diagram of the ball during the brief time interval that it is in contact with the floor. Show 4 or 5 frames as the ball compresses, then another 4 or 5 frame as it expands. What is the direction of $\vec { a }$ during each of these parts of the motion? b. Draw a picture of the ball in contact with the floor and identify all forces acting on the ball. c. Draw a free-body diagram of the ball during its contact with the ground. Is there a net force acting on the ball? If so, in which direction? d. During contact, is the force of the ground on the ball larger than, smaller than, or equal to the weight of the ball? Use your answers to parts a-c to explain your reasoning.

A projectile is fired with initial speed $v_{0}$ feet per second from a height of h feet at an angle of $\theta$ above the horizontal. Assuming that the only force acting on the object is gravity, find the maximum altitude, horizontal range and speed at impact. $v_{0}=80, h=6, \theta=\frac{\pi}{4}$

In outer space, far from other objects, block 1 of mass 40 kg is at position $\langle 7,11,0\rangle \mathrm{m},$ and block 2 of mass 1000 kg is at position $\langle 18,11,0\rangle \mathrm{m}.$ What is the (vector) gravitational force acting on block 2 due to block 1? It helps to make a sketch of the situation.

A simple log bridge in a remote area consists of two parallel logs with planks across them. The logs are Douglas fir with average diameter 300 mm. A truck moves slowly across the bridge, which spans 2.5 m. Assume that the weight of the truck is equally distributed between the two logs. Because the wheelbase of the truck is greater than 2.5 m, only one set of wheels is on the bridge at a time. Thus, the wheel load on one log is equivalent to a concentrated load W acting at any position along the span. In addition, the weight of one log and the planks it supports is equivalent to a uniform load of $850\ \mathrm{N} / \mathrm{m}$ acting on the log. Determine the maximum permissible wheel load W based upon (a) an allowable bending stress of 7.0 MPa, and (b) an allowable shear stress of 0.75 MPa.

A person is driving his motorcycle at a speed of 110 km/h in air at $20^{\circ} \mathrm{C}$. The frontal area of the motorcycle and driver is 0.75 $m^2$. If the drag coefficient under these conditions is estimated to be 0.90, the drag force acting on the car in the flow direction is (a) 379 N(b) 220 N(c) 283 N(d) 308 N(e) 450 N

Consider a 6-m-diameter spherical gate holding a body of water whose height is equal to the diameter of the gate. Atmospheric pressure acts on both sides of the gate. The horizontal component of the hydrostatic force acting on this curved surface is (a) 709 kN(b) 832 kN(c) 848 kN(d) 972 kN(e) 1124 kN