**Problem 1**

Two horses pull horizontally on ropes attached to a stump. The two forces \(\texttip{\vec {F_1}}{F_1_vec}\) = 1230\({\rm N}\) and \(\vec{F}_2\) that they apply to the stump are such that the net (resultant) force \(\texttip{\vec{R}}{R_vec}\) has a magnitude equal to that of \(\vec{F}_1\) and makes an angle of 90\(^\circ\) with \(\vec{F}_1\).

Part A

Find the magnitude of \(\texttip{\vec{F}_{\rm 2}}{F_2_vec}\).

Part B

Find the direction of \(\texttip{\vec{F}_{\rm 2}}{F_2_vec}\) (relative to \(\texttip{\vec {F_1}}{F_1_vec}\)).

**Exercise 1.1**

A dockworker applies a constant horizontal force of 84.0\({\rm N}\) to a block of ice on a smooth horizontal floor. The frictional force is negligible. The block starts from rest and moves a distance 12.5\({\rm m}\) in a time 4.80\({\rm s}\) .

Part A

What is the mass of the block of ice?

Part B

If the worker stops pushing at the end of 4.80\({\rm s}\) , how far does the block move in the next 5.90\({\rm s}\) ?

**Exercise 1.2**

At the surface of Jupiter's moon Io, the acceleration due to gravity is 1.81\({\rm m/s^2}\) . A watermelon has a weight of 40.0\({\rm N}\) at the surface of the earth. In this problem, use 9.81\({\rm m/s^2}\) for the acceleration due to gravity on earth.

Part A

What is its mass on the earth's surface?

Part B

What is its mass on the surface of Io?

Part C

What is its weight on the surface of Io?

**Exercise 1.3**

A chair of mass 14.5\({\rm kg}\) is sitting on the horizontal floor; the floor is not frictionless. You push on the chair with a force \(\texttip{F}{F}\) = 39.0\({\rm N}\) that is directed at an angle of 42.0\({\rm ^\circ}\) below the horizontal and the chair slides along the floor.

Part A

Draw a clearly labelled free-body diagram for the chair.

Draw the vectors starting at the black dot. The location and orientation of the vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded.

Part B

Use your diagram and Newton's laws to calculate the normal force that the floor exerts on the chair.

**Problem 2**

A parachutist relies on air resistance (mainly on her parachute) to decrease her downward velocity. She and her parachute have a mass of 57.5\({\rm kg}\) and air resistance exerts a total upward force of 690\({\rm N}\) on her and her parachute.

Part A

What is the combined weight of the parachutist and her parachute?

Part B

Draw a free-body diagram for the parachutist.

Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded.

Part C

Calculate the net force on the parachutist.

Part D

Is the net force upward or downward?

Part E

What is the magnitude of the acceleration of the parachutist?

Part F

What is the direction of the acceleration?

**Problem 3**

An athlete whose mass is 97.0\({\rm kg}\) is performing weight-lifting exercises. Starting from the rest position, he lifts, with constant acceleration, a barbell that weighs 410\({\rm N}\) . He lifts the barbell a distance of 0.65\({\rm m}\) in a time of 2.0\({\rm s}\) .

Part A

Draw a clearly labelled free-body force diagram for the barbell.

Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded.

Part B

Draw a clearly labelled free-body force diagram for the athlete

Draw the force vectors with their tails at the dot. The orientation of your vectors will be graded. The exact length of your vectors will not be graded but the relative length of one to the other will be graded.

Part C

Use the diagrams in parts A and B and Newton's laws to find the total force that his feet exert on the ground as he lifts the barbell.

Express your answer using two significant figures.

**Exercise 4**

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Figure 1) . The rope will break if the tension in it exceeds 2.35×104\({\rm N}\) , and our hero's mass is 92.4\({\rm kg}\) .

Part A

If the angle between the rope and the horizontal is \(\texttip{\theta }{theta}\) = 11.8\({\rm ^\circ}\), find the tension in the rope.

Part B

What is the smallest value the angle \(\theta\) can have if the rope is not to break?

**Exercise 5**

A man pushes on a piano with mass 160\({\rm {\rm kg}}\) so that it slides at constant velocity down a ramp that is inclined at 13.9\({\rm ^\circ}\) above the horizontal floor. Neglect any friction acting on the piano.

Part A

Calculate the magnitude of the force applied by the man if he pushes parallel to the incline.

Part B

Calculate the magnitude of the force applied by the man if he pushes parallel to the floor.

**Exercise 6**

A 79.0-\({\rm kg}\) painter climbs a ladder that is 2.75\({\rm m}\) long leaning against a vertical wall. The ladder makes an 32.0\({\rm ^\circ}\) angle with the wall.

Part A

How much work does gravity do on the painter?

Part B

Does the answer to part A depend on whether the painter climbs at constant speed or accelerates up the ladder?

**Exercise 7**

Use the work-energy theorem to solve each of these problems. You can use Newton's laws to check your answers. Neglect air resistance in all cases.

Part A

A branch falls from the top of a 86.0 \(\rm m\) tall Australian cedar, starting from rest. How fast is it moving when it reaches the ground?

Part B

A volcano ejects a boulder directly upward 521\({\rm m}\) into the air. How fast was the boulder moving just as it left the volcano?

Part C

A skier moving at 5.00\({\rm m/s}\) encounters a long, rough horizontal patch of snow having coefficient of kinetic friction 0.220 with her skis. How far does she travel on this patch before stopping?

Part D

Suppose the rough patch in part C was only 2.90\({\rm m}\) long? How fast would the skier be moving when she reached the end of the patch?

Part E

At the base of a frictionless icy hill that rises at 21.0\({\rm ^\circ}\) above the horizontal, a toboggan has a speed of 12.0 \({\rm{ m/s}}\) toward the hill. How high vertically above the base will it go before stopping?

**Exercise 8**

A little red wagon with mass 6.80\({\rm kg}\) moves in a straight line on a frictionless horizontal surface. It has an initial speed of 3.40\({\rm m/s}\) and then is pushed 4.3\({\rm m}\) in the direction of the initial velocity by a force with a magnitude of 10.0 \({\rm N}\).

Part A

Use the work-energy theorem to calculate the wagon's final speed.

Express your answer using two significant figures.

Part B

Calculate the acceleration produced by the force.

Express your answer using two significant figures.

**Exercise 9**

A child applies a force \(\vec F\) parallel to the \(x\) -axis to a 10.0-\({\rm kg}\) sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the \(x\) -component of the force she applies varies with the \(x\) -coordinate of the sled as shown in the figure (Figure 1) .

Part A

Calculate the work done by the force \(\vec F\) when the sled moves from \(x\)=0 to \(x\)=8.0\({\rm{ m}}.\)

Express your answer using two significant figurs.

Part B

Calculate the work done by the force \(\vec F\) when the sled moves from \(x\)=8.0\({\rm{ m}}\) to \(x\) =12.0\({\rm{ m}}\).

Express your answer using two significant figurs.

Part C

Calculate the work done by the force \(\vec F\) when the sled moves from \(x\)=0 to \(x\) =12.0\({\rm{ m}}\).

.

Express your answer using two significant figurs.

**Exercise 10**

A tandem (two-person) bicycle team must overcome a force of 175\({\rm N}\) to maintain a speed of 9.00\({\rm m/s}\) .

Part A

Find the power required per rider, assuming that each contributes equally.Express your answer in watts.

Part B

Express your answer in horsepower.

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