A small sphere with mass m is attached to a massless rod of length L that is pivoted at the top, forming a simple pendulum. The pendulum is pulled to one side so that the rod is at an angle θ from the vertical, and released from rest. At this point, what is the linear speed of the sphere? Express your answer in terms of g, θ, L.

of k = 700 N/m. A block of mass m = 1.10 kg is attached to the spring and rests on a frictionless, horizontal surface as in the figure below. The left end of a horizontal spring is attached to a vertical wall, and the right end is attached to a block of mass m. The spring has force constant k. Three positions are labeled along the spring, and the block is pulled to the rightmost position, stretching the spring. The leftmost position, the equilibrium position, is labeled x = 0. The middle position, halfway between the leftmost and rightmost positions, is labeled x = xi⁄2. The rightmost position is labeled x = xi. (a) The block is pulled to a position xi = 6.80 cm from equilibrium and released. Find the potential energy stored in the spring when the block is 6.80 cm from equilibrium. J(b) Find the speed of the block as it passes through the equilibrium position. m/s(c) What is the speed of the block when it is at a position xi/2 = 3.40 cm? m/s

An amusement park ride spins its rider in a circle with a radius of 2.0 m.  Determine the minimum frequency of rotation of the circle that would keep the rider from falling if the friction coefficient between the rider and the wall is 0.68.  Express your answer in units of Hz (cycles per second).  Use g=10 N/kg.  Hint:  The required friction force is the weight of the person.  The normal force of the wall on the rider causes the friction.

A challenge gaining prominence throughout the planet is the increased need for "green" or sustainable energy. In certain parts of the country, wind farms are a viable alternative to conventional energy sources. An average wind turbine, the device which converts rotational motion to electricity, has a maximum capacity of 2.00 MW . However, the wind turbine realistically never runs at full power. If a wind turbine runs at 31.0% capacity for 1.00 year , how much energy does the wind turbine generate?

To form a pendulum, a 0.092 kg ball is attached to one end of a rod of length 0.62 m and negligible mass, and the other end of the rod is mounted on a pivot. The rod is rotated until it is straight up, and then it is released from rest so that it swings down around the pivot. When the ball reaches its lowest point, what are (a) its speed and (b) the tension in the rod? Next, the rod is rotated until it is horizontal, and then it is again released from rest. (c) At what angle from the vertical does the tension in the rod equal the weight of the ball? (d) If the mass of the ball is increased, does the answer to (c) increase, decrease, or remain the same?

A mass m = 0.55 kg is pushed against a spring with spring constant k and held in place with a catch. The spring compresses a distance x = 0.22 m. When the catch is removed, the mass leaves the spring and slides along a frictionless circular loop of radius R = 0.20 m. When the mass reaches the top of the loop, the normal force is equal to three times the weight of the mass. (a) What is the velocity of the mass at point A? (b) What are the magnitude and direction of the net force at point A?

A point-like mass m is constrained to move along the z-axis only. It is attached to two springs, each of spring constant k, and each of rest length 0. One spring has one of its ends fixed at x = 0 and y = h; the other spring is fixed at one end to z = 0 and y=-h as shown in the figure below. Gravity does not act on the system. The springs do not bend but contract and extend along straight lines between their end points. y m (d) Determine the equilibrium position of the system and the frequencies of small oscillations about this equilibrium position. Hint: use approximations (i.e. Taylor expansion) to simplify the equation of motion near the equilibrium position to that of a harmonic oscillator. (e) Repeat part (d) for the case where the two springs have different spring constants k₁ and k₂ and are placed at different distances on the y-axis h₁ and h₂.

The wheel is attached to the spring. The mass of the wheel is m=20 kg. The radius of the wheel is 0.6m. The radius of gyration KG=0.4 m. The spring's unstretched length is Lo=1.0 m. The stiffness coefficient of the spring is k=2.0 N/m. The wheel is released from rest at the state 1 when the angle between the spring and the vertical direction is 8-30°. The wheel rolls without slipping and passes the position at the state 2 when the angle is 8=0°. The spring's length at the state 2 is L2=4 m. (1) If the mass center G is set as the origin (datum), the gravitational potential energy at the state 1 is_____ (two decimal places) 1116441 L₂ State 2 State 1

The figure blow shows the kinetic energy of a simple pendulum versus its angle θ from the vertical.  Note the energy units are in milli-joules.  The vertical axis is set by Ks=4mJ and the mass of the pendulum bob is 0.200 kg.  Determine the length of the pendulum in meters.  Use g = 9.8 N/kg.

A gymnast swings on the high bar as shown in the figure. Starting from rest directly over the bar, he swings around the bar while keeping his arms and legs outstretched. Treating the gymnast as though his entire mass were concentrated at a point y=0.87 m from the bar, determine his speed as he passes under the bar at position A. Note that you do not need know his mass to solve this problem.   It is asking for the units in m/s