Review. A particle moves along the x axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration –0.320 m/s2. Suppose it moves as a particle under constant acceleration for 4.50 s. Find (a) its position and (b) its velocity at the end of this time interval. Next, assume it moves as a particle in simple harmonic motion for 4.50 s and x = 0 is its equilibrium position. Find (c) its position and (d) its velocity at the end of this time interval.

A particle moves along the x axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration $-0.320 \mathrm{m} / \mathrm{s}^{2}$. Suppose it moves as a particle under constant acceleration for 4.50 s. Find (a) its position and (b) its velocity at the end of this time interval. Next, assume it moves as a particle in simple harmonic motion for 4.50 s and x = 0 is its equilibrium position. Find (c) its position and (d) its velocity at the end of this time interval.

Your forehead can withstand a force of about 6.0 kN before fracturing, while your cheekbone can only withstand about 1.3 kN. a. If a 140 g baseball strikes your head at 30 m/s and stops in 0.0015 s, what is the magnitude of the ball's acceleration? b. What is the magnitude of the force that stops the baseball? c. What force does the baseball apply to your head? Explain. d. Are you in danger of a fracture if the ball hits you in the forehead? In the cheek?

The position of a particle moving along x-axis is given by x(t) = 4.0 - 2.0t m. (a) At what time does the particle cross the origin? (b) What is the displacement of the particle between t = 3.0 s and t = 6.0 s?

(a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as x = 0. The object is pulled down an additional 18.0 cm and released from rest to oscillate without friction. What is its position x at a moment 84.4 s later? (b) Find the distance traveled by the vibrating object in part (a). (c) What If? Another hanging spring stretches by 35.5 cm when an object of mass 440 g is hung on it at rest. We define this new position as x = 0. This object is also pulled down an additional 18.0 cm and released from rest to oscillate without friction. Find its position 84.4 s later. (d) Find the distance traveled by the object in part (c). (e) Why are the answers to parts (a) and (c) so different when the initial data in parts (a) and (c) are so similar and the answers to parts (b) and (d) are relatively close? Does this circumstance reveal a fundamental difficulty in calculating the future?

Pat always trips on the third step and she thinks that step may be a different size. The contractor told her that all the treads and risers are perpendicular to each other. Is that enough information to state that the steps are the same size? In other words, if $\overline{WN}\perp\overline{NZ}$ and $\overline{ZH}\perp\overline{HK}$, is $\triangle WNZ\cong\triangle ZHK$?

“The wave” is a particular type of pulse that can propagate through a large crowd gathered at a sports arena. The elements of the medium are the spectators, with zero position corresponding to their being seated and maximum position corresponding to their standing and raising their arms. When a large fraction of the spectators participates in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on people’s reaction time, which is typically on the order of 0.1 s. Estimate the order of magnitude, in minutes, of the time interval required for such a pulse to make one circuit around a large sports stadium. State the quantities you measure or estimate and their values.

The 20-lb block and the 50-lb block are connected by a rope hung over the fixed curved surface. If the system is on the verge of slipping, calculate the coefficient of friction between the rope and the surface.

A totally reflecting disk has radius 8.00 $\mu m$ and average density 600 $\mathrm{kg} / \mathrm{m}^{3}$. A laser has an average power output $P_{\mathrm{av}}$ spread uniformly over a cylindrical beam of radius 2.00 mm. When the laser beam shines upward on the disk in a direction perpendicular to its flat surface, the radiation pressure produces a force equal to the weight of the disk. What value of $P_{\mathrm{av}}$ is required?

A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Find an expression for the position of the particle as a function of time. Determine (b) the maximum speed of the particle and (c) the earliest time (t > 0) at which the particle has this speed. Find (d) the maximum positive acceleration of the particle and (e) the earliest time (t > 0) at which the particle has this acceleration. (f) Find the total distance traveled by the particle between t = 0 and t = 1.00 s.

The position of a particle is given by the expression x = 4.00 cos ($3.00\pi t + \pi$), where x is in meters and t is in seconds. Determine (a) the frequency and (b) period of the motion, (c) the amplitude of the motion, (d) the phase constant, and (e) the position of the particle at t = 0.250 s.

The amplitude of a system moving in simple harmonic motion is doubled. Determine the change in (a) the total energy, (b) the maximum speed, (c) the maximum acceleration, and (d) the period.

The mass $A$ starts from rest at $t=0$ and falls with a constant acceleration of $8 \mathrm{~m} / \mathrm{s}^2$. When the mass has fallen one meter, determine the magnitudes of (a) the angular velocity of the pulley and (b) the tangential and normal components of acceleration of a point at the outer edge of the pulley.

An engineer is designing a triangular weir for measuring the flow rate of a stream of water that has a discharge of 6 cfm. The weir has an included angle of 45$^{\circ}$ and a coefficient of discharge of 0.6. Find the head on the weir.