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Three forces of equal magnitude are applied to a \(3 \, \text{m} \times 2 \, \text{m}\) rectangle. Force \(F_1\) and \(F_2\) act at \(45^\circ\) angles to the vertical as shown, while \(F_3\) acts horizontally.
In short:
\(F_1\): applied at \((0, -2)\), direction SW \(45^\circ\)
\(F_2\): applied at \((2, -2)\), direction NW \(45^\circ\)
\(F_3\): applied at \((3, -1)\), direction east
Points of rotation: \(A = (0, 0)\), \(B = (0, -1)\), \(C = (1, -1)\)
A typical \( 68 \)\(-\text{kg} \) person can maintain a steady energy expenditure of \( 480 \) \( \text{W} \) on a bicycle. Approximately how many Calories are “burned” when the person rides a bicycle for \( 15 \) minutes? A typical energy efficiency for the human body is \( 25\% \), which takes into account the release of thermal energy.
A linear spring of force constant \( k \) is used in a physics lab experiment. A block of mass \( m \) is attached to the spring and the resulting frequency, \( f \), of the simple harmonic oscillations is measured. Blocks of various masses are used in different trials, and in each case, the corresponding frequency is measured and recorded. If \( f^{2} \) is plotted versus \( \frac{1}{m} \), the graph will be a straight line with slope
A ring is pulled on by three forces. If the ring is not moving, how big is the force \( F \)?
Two boxes are tied together by a string and are sitting at rest on a frictionless surface. Between the two boxes is a massless compressed spring. The string trying the two boxes is then cut and the spring expands, pushing the boxes apart. The box on the left has four times the mass of the box on the right.
Why do you tend to slide across the car seat when the car makes a sharp turn?
In which of the following cases does a car have a negative velocity and a positive acceleration? A car that is traveling in the:
A train is moving to the right at \( 20 \) \( \text{m/s} \). A passenger on the train throws a ball horizontally to the left at \( 5 \) \( \text{m/s} \) (relative to the train).
A person’s center of mass is easily found by having the person lie on a reaction board. A horizontal, \( 2.3 \) \( \text{m} \)-long, \( 6.1 \) \( \text{kg} \) reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A \( 64 \) \( \text{kg} \) woman lies on the reaction board with her feet over the pivot. The scale reads \( 27 \) \( \text{kg} \). What is the distance from the woman’s feet to her center of mass? Express your answer with the appropriate units.
A \(4 \, \text{kg}\) mass is traveling at \(10 \, \text{m/s}\) to the right when it collides elastically with a stationary \(7 \, \text{kg}\) mass. The \(7 \, \text{kg}\) mass then travels at \(2 \, \text{m/s}\) at an angle of \(22^\circ\) below the horizontal. What is the velocity of the \(4 \, \text{kg}\) mass?
An airplane is traveling \( 900. \) \( \text{km/h} \) in a direction \( 38.5^\circ \) west of north.
An airplane with a speed of \( 97.5 \, \text{m/s} \) is climbing upward at an angle of \( 50.0^\circ \) with respect to the horizontal. When the plane’s altitude is \( 732 \, \text{m} \), the pilot releases a package.
The heaviest train ever pulled by a single engine was over \( 2 \, \text{km} \) long. A force of \( 1.13 \times 10^5 \, \text{N} \) is needed to get the train to start moving. If the coefficient of static friction is \( 0.741 \) and the coefficient of kinetic friction is \( .592 \), what is the train’s mass?
A \(1 \, \text{kg}\) mass and an unknown mass \(M\) hang on opposite sides of a pulley suspended from the ceiling. When the masses are released, \(M\) accelerates downward at \(5 \, \text{m/s}^2\). Find the value of \(M\).
Consider the following cases of inelastic collisions.
Case (1) – A car moving at \(75 \, \text{mph}\) collides with another car of equal mass moving at \(75 \, \text{mph}\) in the opposite direction and comes to a stop.
Case (2) A car moving at \(75 \, \text{mph}\) hits a stationary steel wall and rolls back.
The collision time is the same for both cases. In which of these cases would result in the greatest impact force?
A merry-go-round spins freely when Diego moves quickly to the center along a radius of the merry-go-round. As he does this, it is true to say that
Two points, \( A \) and \( B \), are in a pipe carrying a flowing ideal fluid. Point \( B \) is \( 2.0 \) \( \text{m} \) higher than point \( A \), and the fluid speed at \( B \) is twice the speed at \( A \). If the pressure at \( A \) is \( P_A \), which of the following expressions correctly represents the pressure at \( B \) \( (P_B) \)?
A skateboarder, with an initial speed of \( 20.0 \, \text{m/s} \), rolls to the end of friction-free incline of length \( 25 \, \text{m} \). At what angle is the incline oriented above the horizontal?
An ideal fluid flows through a pipe with radius \( Q \) and flow speed \( V \). If the pipe splits up into three separate paths, each with radius \( \frac{Q}{2} \), what is the flow speed through each of the paths?
A train consists of \(50\) cars, each of which has a mass of \(6.1 \times 10^{3} \, \text{kg}\). The train has an acceleration of \(8.0 \times 10^{-2} \, \text{m/s}^2\). Ignore friction and determine the tension in the coupling at the following places:
A \( 1.5 \; \text{kg} \) mass attached to a spring with a force constant of \( 20.0 \; \text{N/m} \) oscillates on a horizontal, frictionless track. At \( t = 0 \), the mass is released from rest at \( x = 10.0 \; \text{cm} \). (That is, the spring is stretched by \( 10.00 \; \text{cm} \).)
Three identical rocks are launched with identical speeds from the top of a platform of height \( h_0 \).
Which of the following correctly relates the magnitude \( v_y \) of the vertical component of the velocity of each rock immediately before it hits the ground?
A typical \( 68 \text{-kg} \) person generates a steady mechanical power output of \( 120 \text{ W} \) at the pedals of a bicycle. Approximately how many Calories are “burned” (total metabolic energy expended) when the person rides a bicycle for \( 15 \text{ minutes} \)? A typical energy efficiency for the human body is \( 25\% \), which takes into account the release of thermal energy. Note (\( 1 \text{ Cal} = 4186 \text{ J} \)).
A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of 10.8 N·m which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?
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