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.

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?

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?

An element in biaxial stress is subjected to stresses $\sigma_{x}=6250$ psi and $\sigma_{y}=-1750$ psi. Using Mohr’s circle, determine: (a) The stresses acting on an element oriented at a counterclockwise angle $\theta=55^{\circ}$ from the x axis. (b) The maximum shear stresses and associated normal stresses. Show all results on sketches of properly oriented elements.

The impulse J due to a force F is the product of the force times the amount of time t for which the force acts. When the force varies over time, $J=\int_{t_{1}}^{t_{2}} F(t) d t$. We can model the force acting on a rocket during launch by an exponential function $F(t)=A e^{b t}$, where A and b are constants that depend on the characteristics of the engine. At the instant lift-oft occurs (t=0), the force must equal the weight of the rocket. (a) Suppose the rocket weighs 25,000 N (a mass of about 2500 kg or a weight of 5500 lb), and 30 seconds after lift-off the force acting on the rocket equals twice the weight of the rocket. Find A and b. (b) Find the impulse delivered to the rocket during the first 30 seconds after the launch.

A windmill has an initial angular momentum of $8500 \mathrm{~kg} \cdot \mathrm{m}^2 / \mathrm{s}$. The wind picks up, and $5.86 \mathrm{~s}$ later the windmill's angular momentum is $9700 \mathrm{~kg} \cdot \mathrm{m}^2 / \mathrm{s}$. What was the torque acting on the windmill, assuming it was constant during this time?

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

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.

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.

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 $u=g-k v$.]

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 vertical component of the hydrostatic force acting on this curved surface is (a) 89 kN(b) 270 kN(c) 327 kN(d) 416 kN(e) 505 kN

In a common but dangerous prank, a chair is pulled away as a person is moving downward to sit on it, causing the victim to land hard on the floor. Suppose the victim falls by 0.50 meter, the mass that moves downward is 70 kilogram, and the collision on the floor lasts 0.082 second. What is the magnitudes of the impulse acting on the victim from the floor during the collision?

A long, straight wire containing a semicircular region of radius $0.95 \mathrm{~m}$ is placed in a uniform magnetic field of magnitude $2.20 \mathrm{~T}$. What is the net magnetic force acting on the wire when it carries a current of $3.40 \mathrm{~A}$? (what does symmetry tell you about the forces on the upper and lower halves of the semicircular region?)

The force F per unit length of a conducting wire carrying a current I in a magnetic field B is F = I × B. Find the force acting on a circuit whose shape is given by x = sin θ, y = cos θ, $z=\sin \frac{1}{2} \theta$, when current I flows in it and when it lies in a magnetic field B = xi − yj + k.