Example 8.1 Ball in Free Fall A ball of mass m is dropped from a height h above the ground. (A) Neglecting air resistance, determine the speed of the ball when it is at a height y above the ground. Choose the system as the ball and the Earth. (B) Find the speed of the ball again at height y by choosing the ball as the system.
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Example 8.1 Ball in Free Fall
A ball of mass m is dropped from a height h above the ground.
(A) Neglecting air resistance, determine the speed of the ball when it is at a height y above the ground. Choose the system as the ball and the Earth.
(B) Find the speed of the ball again at height y bychoosing the ball as the system.
Example 8.2 A Grand Entrance
You are designing an apparatus tosupport an actor of mass 65 kg whois to “fly” down to the stage duringthe performance of a play. You attachthe actor’s harness to a 130-kgsandbag by means of a lightweightsteel cable running smoothly overtwo frictionless pulleys. You need3.0 m of cable between the harnessand the nearest pulley so that thepulley can be hidden behind acurtain. For the apparatus to worksuccessfully, the sandbag must never lift above the floor as the actor swingsfrom above the stage to the floor. Let us call the initial angle that the actor’scable makes with the vertical θ.What is the maximum value θ can have before the sandbag lifts off the floor?
Example 8.3 The Spring-Loaded Popgun
The launching mechanism of a popgun consists of a trigger-released spring. Thespring is compressed to a position yA, and the trigger is fired. The projectile of mass m rises to a position yC above the position at which it leaves the spring, indicated as position yB = 0. Consider a firing of the gun for which m = 35.0 g, yA = –0.120 m, and yC = 20.0 m.
(A) Neglecting all resistive forces, determine the spring constant.
(B) Find the speed of the projectile as it moves through theequilibrium position B of the spring.
Example 8.4 A Block Pulled on a Rough Surface
A 6.0-kg block initially at rest is pulled to the right along a horizontal surface by a constant horizontal force of 12 N.
(A) Find the speed of the block after it has moved 3.0 m if the surfaces in contact have a coefficient of kinetic friction of 0.15.
(B) Suppose the force F is applied at an angle θ. At what angle should the forcebe applied to achieve the largest possible speed after the block has moved 3.0 mto the right?
Conceptual Example 8.5 Useful Physics for Safer Driving
A car traveling at an initial speed v slides a distance d to a halt after its brakes lock. If the car’s initial speed is instead 2v at the moment the brakes lock, estimate the distance it slides.
Example 8.6 A Block–Spring System
A block of mass 1.6 kg is attached to a horizontal spring that has a force constant of 1000 N/m. The spring is compressed 2.0 cm and is then released from rest.
(A) Calculate the speed of the block as it passes through the equilibrium position x = 0 if the surface is frictionless.
(B) Calculate the speed of the block as it passes through the equilibrium position if a constant friction force of 4.0 N retards its motion from the momentit is released.
Example 8.7 Crate Sliding Down a Ramp
A 3.00-kg crate slides down a ramp. The ramp is 1.00 m in length and inclined at an angle of 30.0º. The crate starts from rest at the top, experiences a constant friction force of magnitude 5.00 N, and continues to move a short distance on the horizontal floor after it leaves the ramp.
(A) Use energy methods to determine the speed of the crate at the bottom of theramp.
(B) How far does the crate slideon the horizontal floor if itcontinues to experience a frictionforce of magnitude 5 N?
Example 8.8 Block–Spring Collision
A block having a mass of 0.80 kg is given an initial velocity vA = 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in the figure.
(A) Assuming the surface to be frictionless,calculate the maximum compression of thespring after the collision.
(B) Suppose a constant force of kineticfriction acts between the block and thesurface, with μk = 0.50. If the speed of theblock at the moment it collides with thespring is vA = 1.2 m/s, what is the maximumcompression xC in the spring?
Example 8.9 Connected Blocks in Motion
Two blocks are connected by a light string that passes over a frictionless pulley as shown in the figure. The block of mass m1 lies on a horizontal surface and is connected to a spring of force constant k. The system is released from rest whenthe spring is unstretched. If the hanging block of mass m2 falls a distance h before coming to rest, calculate the coefficient of kinetic friction between the block of mass m1 and the surface.
Conceptual Example 8.10 Interpreting the Energy Bars
The energy bar charts in the right figure show three instants in the motion of thesystem in the left figure and described in Example 8.9. For each bar chart, identify the configuration of the system that corresponds to the chart.
Example 8.11 Power Delivered by an Elevator Motor
An elevator car has a mass of 1600 kg and is carrying passengers having a combined mass of 200 kg. A constant friction force of 4000 N retards its motion.
(A) How much power must a motor deliver to lift the elevator car and its passengers at a constant speed of 3.00 m/s?
(B) What power must the motor deliver at the instantthe speed of the elevator is v if the motor is designedto provide the elevator car with an upward accelerationof 1.00 m/s²?
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