6.8: Chapter 6 Homework Problems

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Exercise \(\PageIndex{1}\)

A boy is pulling a sled full of snowballs weighing 30 lbs across a snowy flat surface (\(\mu_s = 0.3, \, \mu_k\) = 0.1). Find the force \(F\) needed to keep the sled moving at a constant speed.

A sled faces left on a flat surface. A pulling force is applied at the front of the sled, pointing left and up at 30 degrees above the horizontal. — Figure \(\PageIndex{1}\): problem diagram for Exercise \(\PageIndex{1}\). A sled is pulled across a flat surface with a force applied at an angle.

Solution: \(F_{pull} = 3.28\) lbs

Exercise \(\PageIndex{2}\)

A wooden box sits on a concrete slope (\(\mu_s = 0.62, \, \mu_k\) = 0.55). How much force would be needed to start pulling this box up the ramp? If we let go of the box, would it slide down the ramp?

A 60-kg box sits on a 25-degree incline, and experiences a pulling force on its right side with a direction parallel to the incline. — Figure \(\PageIndex{2}\): problem diagram for Exercise \(\PageIndex{2}\). A box is pulled up on a 25° incline by a force applied parallel to the ramp.

Solution

\(F_{pull} = 578.9\) N

Box will not slip if released.

Exercise \(\PageIndex{3}\)

A wheelbarrow with a weight of 60 lbs and the dimensions shown below sits on a ten-degree incline. Assume friction exists at the rear support (A) but no friction exists at the wheel (B). What is the minimum coefficient of friction needed between the support and the ground to keep the wheelbarrow from sliding down the hill?

Side view of a wheelbarrow, with one front wheel whose point of contact with the ground is B and one rear support whose point of contact with the ground is A, facing uphill. The center of mass is 24 inches in front of A, 12 inches behind B, and 18 inches above the ground. — Figure \(\PageIndex{3}\): problem diagram for Exercise \(\PageIndex{3}\). A 60-lb wheelbarrow faces uphill on a 10° incline and experiences friction between the ground and its rear support.

Solution: \(\mu_s = 0.418\)

Exercise \(\PageIndex{4}\)

The car below weighs a total of 1500 lbs, has a center of mass at the location shown, and is rear-wheel drive (only the rear wheels will create a friction force). Assuming that the tires are rubber and the surface is concrete \((\mu_s\) = 0.9), what is the maximum angle of the hill \((\theta)\) that the car will be able to climb at a constant rate before the wheels start to slip? What is the maximum angle if the car is front-wheel drive?

Side view of a car facing uphill on an incline of angle theta. Its wheels are 8 feet apart, and its center of mass is 3 feet behind the front wheel and 1.5 feet above the ground. — Figure \(\PageIndex{4}\): problem diagram for Exercise \(\PageIndex{4}\). A 1500-lb car climbs uphill at a constant rate on an incline of angle \(\theta\).

Solution

\(\theta_{max} = 22.0\)° for rear-wheel drive

\(\theta_{max} = 25.7\)° for front-wheel drive

Exercise \(\PageIndex{5}\)

The fridge shown below has a total weight of 120 lbs and a center of mass as shown below. The fridge is pushed as shown until it either starts to slide or tips over. What is the minimum coefficient of friction needed for the fridge to tip before it starts sliding?

Front view of a fridge on a level surface, with center of mass located 1.5 feet from the left and right sides of the fridge and 3 feet above the ground. A pushing force is applied on its left side, applied 2 feet above the ground. — Figure \(\PageIndex{5}\): problem diagram for Exercise \(\PageIndex{5}\). A 120-lb fridge on a level surface experiences a pushing force towards the right.

Solution: \(\mu_s = 0.75\) at a minimum

Exercise \(\PageIndex{6}\)

You have a bookcase with the dimensions and weight shown below. You are examining the safety of your design.

If a toddler were to pull on the bookcase as shown, what is the pulling force that would tip it over? (assume the center of gravity is the center of the bookcase and there is no slipping)
What would the static coefficient of friction need to be to have the case slide before it tips over?

Right: photograph of a bookcase with a rectangular casing and horizontal shelves in the interior. Left: diagram of side view of said bookcase, with a weight of 120 lbs, width 1 foot and height 5 feet. A pulling force directed downwards and to the left (at 30 degrees below the horizontal) is applied to the left side of the case, 2 feet above the ground. — Figure \(\PageIndex{6}\): problem diagram for Exercise \(\PageIndex{6}\). A 120-lb bookcase on a level surface experiences a downwards pulling force applied at an angle on its left side.

Solution

\(F_{pull} = 34.64\) lbs

\(\mu_s = 0.218\) at a maximum

Exercise \(\PageIndex{7}\)

The wedge shown below is pressed by a log splitter into a log with a force of 200 lbs. Assuming the coefficient of friction (both static and kinetic) between the steel wedge and the wood of the log is 0.3, what is the magnitude of the normal force exerted on either side of the log?

A wedge with an 8-degree point angle is pressed vertically, point-down, into a wood rectangle with a force of 200 lbs. — Figure \(\PageIndex{7}\): problem diagram for Exercise \(\PageIndex{7}\). A wedge with a point angle of 8° is pressed point-down into a log.

Solution: \(F_{N1} = F_{N2} = 271.0\) lbs

Exercise \(\PageIndex{8}\)

The power screw in the screw jack shown below has an outside diameter of one and a half inches and a total of three threads per inch. Assume the coefficients of friction are both 0.16.

What is the moment required to create a two-ton (4000 lb) lifting force?
Is this power screw setup self-locking?

A screw jack with its base on a flat surface experiences a downwards force of 4000 lbs from the load placed on top of it. — Figure \(\PageIndex{8}\): problem diagram for Exercise \(\PageIndex{8}\). A screw jack experiences a downwards force of 4000 lbs from the load placed on it.

Solution

\(M_{lift} = 58.3\) ft-lbs

Screw is self-locking.

Exercise \(\PageIndex{9}\)

The end bearing as shown below is used to support a rotating shaft with a load of 300 N on it. If the shaft and the bearing surface are both lubricated steel (assume the coefficients of friction are both 0.06), what is the moment exerted by the friction forces for…

A solid shaft with a diameter of 2 cm?
A hollow shaft with an outside diameter of 2 cm and an inside diameter of 1.5 cm?

An end bearing supports a shaft that undergoes clockwise rotation, with a 300 N load on it. — Figure \(\PageIndex{9}\): problem diagram for Exercise \(\PageIndex{9}\). An end bearing supports a rotating shaft that experiences a 300 N load.

Solution

\(M_{friction} = 0.12\) N-m (solid shaft)

\(M_{friction} = 0.159\) N-m (hollow shaft)

Exercise \(\PageIndex{10}\)

A 120-lb person is being lifted by a rope thrown over a tree branch as shown below. If the static coefficient of friction between the rope and the tree branch is 0.61, what is the pulling force required to start lifting the person? What is the pulling force required to keep them from falling?

A tree branch, represented end-on as a circle, has a rope thrown over it. The right end of the rope hangs straight down and holds a person; the left end of the rope experiences a pulling force downwards and to the left, at 45 degrees below the horizontal. — Figure \(\PageIndex{10}\): problem diagram for Exercise \(\PageIndex{10}\). A rope thrown over a branch is pulled downwards at an angle at one end, to lift and hold a 120-lb person on the other end.

Solution

\(F_{lift} = 505.1\) lbs

\(F_{hold} = 28.5\) lbs