A car drives at a speed of 61 km/h to a 1 mm thick layer of water on the road and slips. We call this slip, aquaplaning. The contact area between the water and the tires of the car is . Calculate the force with which the layer of water acts on the car. The coefficient of water viscosity is .

Example 2.

A plate with a surface area is slowly dragged along the surface of an unknown liquid. The depth of the fluid is and the plate is moving at a speed of . What is the viscosity of this unknown liquid if the plate is pulled by a force ? Which liquid is it?

An iron ball with a radius of 2 mm falls uniformly in oil. The rate of fall is 2 mm/s. During the fall, the linear law of resistance applies. What is the viscosity of the oil? The densities of iron and oil are and respectively.

A piston moves in a cylinder so that there is a 0.1 mm layer of oil between the piston and the cylinder. The surface of the cylinder jacket is . What is the longitudinal force required for the piston to move at a speed of 20 cm/s? The coefficient of viscosity of the oil is .

Which of the cases described below requires quadratic law of resistance and which requires linear law of resistance?

A bullet fired from a gun travels through the air at a speed of . Its diameter is , the air density is , and the air viscosity is .
A sphere of diameter of moves through honey at a speed of . The density of honey is and its viscosity is .
A soccer ball flies through the air at a speed of . Its radius is .
A cube with a side length of sinks sideways to the bottom of a lake at a speed of . The viscosity of water is .

Joe pulls a sled at an angle to the horizontal surface. The frictional force between the sled and the snow is 42 N. With what force does Joe pull the sled to move uniformly?

A body with mass m is dragged at a constant speed over a rough surface. The force is measured with a force metre (spring balance). The force metre shows a force of 0.6 N. What is the value of the frictional force? If we slightly increase the speed, will the force metre show the same force?

A 200 g box is dragged with a spring balance on a horizontal rough surface so that it moves uniformly. The force shown by the spring balance is 0.3 N.

What is the frictional force?
If another box of the same mass is placed on the previous box, what is the force required for uniform motion now, and what is the friction force?

A rectangular box of lengths of edges:

is dragged on a horizontal rough surface on a side with edges a, b. The mass of the box is 10 kg. If the box is pulled at a constant speed, a force of 20 N is applied. What is the frictional force? What is the frictional force if the box is pulled on a side with edges a, c and b, c.

10.

On the table is a log weighing 2 kg. The log is attached to a string passed over an attached pulley to an unknown weight, as shown in the figure. What is the mass of the unknown weight so that the log will move uniformly? The frictional force between the log and the base is 4 N.

11.

A body slides uniformly down a slope with frictional force of 4 N. What is the dynamic component of the weight? How much force do we need to pull it with to move it uniformly up the slope?

12.

A box of mass 4 kg slides uniformly down a slope at an angle of inclination . What is the frictional force between the box and the slope?

13.

Two blocks of masses 1.5 kg and 1 kg are connected by a string as shown. A rope is attached to the right block and pulled so that both blocks slide at a constant speed v on a flat, rough surface. The frictional force of the right block is 3 N and that of the left 2 N. What is the force in the pulling rope and what is the force in the rope connecting the two blocks?

14.

A box at a uniform speed glides down a slope at an inclination of . The frictional force between the box and the slope is 6 N. What is the mass of the box?

15.

An engine drives a ship at a constant speed with a force of 4 kN.

What is the force of water resistance?
Does the engine power depend on the weight of the cargo on the ship? Explain why.

16.

A crate weighing lies on the ground. A force of is applied to move it.

Draw all the forces acting on the body.
What is the coefficient of friction?
What is the force exerted on the ground by the crate before it begins to move?

17.

A block with a mass of lies on a flat surface. It is acted upon by a force of at an angle of with respect to the horizontal. At a slightly higher force, the body begins to move, what is the coefficient of static friction between the body and the surface?

18.

A wooden log with a mass of is pressed against the wall. How much force do we have to exert on the log to keep it from slipping if the coefficient of static friction between the log and the wall is ?

19.

On the table lies a body with a mass of . The body is connected by a pulley to another body with a mass of . What must be the coefficient of static friction between the table and the first body so that the body does not slip?

20.

On a table lies a body with mass . The body is connected with a string via a pulley to another body of mass . The coefficient of friction between the first body and the table is . With how much force do we have to pull the first body to slide uniformly and the second to lift uniformly?

21.

A log lies in a gutter. The gutter is V-shaped and the angle between the sides of the gutter is . With what force do we have to pull the log in the direction of its axis to make it move uniformly? The coefficient of friction between the log and the gutter is .

22.

Two bodies with masses and are connected with a cord on a horizontal base as shown below.

Mass on string

The coefficient of friction of the first body is and the second is .

What force is needed to move the two bodies uniformly?
What force is needed to pull uniformly with the cord between the bodies?

23.

What is the initial velocity of a body sliding on a rough horizontal surface if it stops at a distance of ? The coefficient of friction between the body and the surface is .

24.

Three bodies of equal mass are connected by a rope and placed on a slope as shown in the figure. The slope angle is . What should be the coefficient of friction between the bodies and the slope so that they do not slip?

25.

A body lies on an inclined plane. The slope of the inclined plane is slowly increased until the body slips. This happens at angle . What is the coefficient of static friction between the body and the plane?

26.

A sled slides uniformly down an inclined plane with a 15 percent slope. What is the coefficient of kinetic friction between the sled and the plane?

27.

A skydiver, who is falling vertically through the air, opens his parachute. Which row describes the velocity of the skydiver immediately after he opens his parachute?

	direction of velocity	magnitude of velocity
A	downwards	decreases
B	downwards	increases
C	upwards	decreases
D	upwards	increases

28.

The graph shows how quantity P varies with quantity Q for a body falling vertically downwards in a uniform gravitational field with air resistance.

Which pair of quantities could be represented by P and Q?

	P	Q
A	acceleration	force of air resistance
B	acceleration	time
C	velocity	force of air resistance
D	velocity	time

29.

A submarine descends vertically at constant velocity. The three forces acting on the submarine are viscous drag, upthrust and weight. Which relationship between their magnitudes is correct?

weight < drag
weight = drag
weight < upthrust
weight > upthrust

30.

A car of mass has a horizontal driving force of acting on it. It has a forward horizontal acceleration of .

What is the resistive force acting horizontally?

31.

A snowflake and a raindrop are in still air. They both fall from rest at the same time and from the same height, far above the ground. The snowflake and raindrop contain the same mass of water. Assume that there is no evaporation or melting. Also assume that, for a given speed, the drag force acting on the snowflake is greater than the drag force acting on the raindrop.

Which statement about the snowflake and raindrop is correct?

The raindrop takes more time than the snowflake to reach terminal velocity.
The raindrop takes more time than the snowflake to reach the ground.
They reach the same terminal velocity.
They take the same amount of time to reach the ground.

32.

A ship of mass is approaching a harbor with speed . By using reverse thrust it can maintain a constant total stopping force of . How long will it take to stop?

33.

A rocket of mass sits on a launch pad on the Earth’s surface. The rocket motors provide an upward force of on the rocket. What is the initial acceleration of the rocket?

34.

An air bubble in a tank of water is rising with terminal (constant) velocity. The forces acting on the bubble are , and , as shown.

The upthrust on the bubble is . Which statement about the forces is correct?

is the viscous force on the bubble, is the weight of the bubble and .
is the viscous force on the bubble, is the weight of the bubble and .
is the weight of the bubble, is the viscous force on the bubble and .
is the weight of the bubble, is the viscous force on the bubble and .

35.

Two balls of identical shape and size but different masses are falling through the same liquid. The sum of the drag force and upthrust acting on each ball is equal to its weight. Which statement about the two balls is correct?

The heavier ball has a larger acceleration than the lighter ball.
The heavier ball has a smaller deceleration than the lighter ball.
The heavier ball is falling at the same speed as the lighter ball.
The heavier ball is falling at a larger speed than the lighter ball.

36.

A box in air slides with increasing speed down a rough slope from point P to point Q.

The slope surface exerts a constant frictional force on the box. As the box moves from P to Q, there are changes to the magnitudes of its acceleration and the total resistive force acting on it. Which row describes the changes?

	magnitude of acceleration	magnitude of total resistive force
A	increases	decreases
B	decreases	decreases
C	increases	increases
D	decreases	increases

37.

The resultant force on a raindrop of mass falling with velocity is given by the equation

Where is a constant and is the acceleration of free fall.

What is the velocity of the raindrop when it reaches a constant (terminal) velocity?

38.

A child on a sledge slides down a hill with acceleration . The hill makes an angle with the horizontal.

The total mass of the child and the sledge is . The acceleration of free fall is . What is the frictional force ?

39.

A ball of mass is thrown vertically into the air. When the ball has speed , the air resistance acting on the ball is . What is the magnitude of the acceleration of the ball when its speed is as it rises and as it falls?

	acceleration when ball is rising	acceleration when ball is falling
A
B
C
D

40.

A snowflake is falling from the sky on a still day. Its weight acts vertically downwards and air resistance acts vertically upwards. As the snowflake falls, air resistance increases until it is equal to the weight and there is no resultant force acting on the snowflake.

When the forces become equal, which statement is correct?

The snowflake accelerates.
The snowflake decelerates.
The snowflake is stationary.
The snowflake moves at a constant velocity.

41.

An ice-hockey puck of mass moves with an initial speed of along the surface of an ice rink. The puck slides a distance of in a straight line before stopping. What is the average frictional force acting on the puck?

42.

A car of mass travels at constant speed up a slope at an angle to the horizontal, as shown in the diagram. Air resistance and friction provide a resistive force . The acceleration of free fall is .

What is the force needed to propel the car at this constant speed?

43.

The drag force acting on a sphere moving through a fluid is given by the expression

where is a constant, is the density of the fluid and is the speed of the sphere.

A ball of weight falls vertically from rest in air. The drag force acting on the ball is given by the expression above. The ball reaches a constant (terminal) speed of . Assume that the upthrust acting on the ball is negligible and that the density of the air is uniform. For the instant when the ball is travelling at a speed of , determine

the drag force on the ball,
the acceleration of the ball.

44.

A ball falls through a liquid at a constant speed. It is acted upon by three forces: an upthrust, a drag-force and its weight. Which statement is correct?

The drag-force increases with increasing depth.
The drag-force is equal to the sum of the upthrust and weight.
The upthrust is constant with increasing depth.
The weight is greater than the sum of the drag-force and the upthrust.

45.

A solid sphere falls at constant (terminal) velocity in a liquid. The three forces acting on the sphere are shown in the diagram.

How are the three forces related?

46.

A body falling in a uniform gravitational field encounters air resistance. The air resistance increases until terminal velocity is reached. Which factor does not affect its terminal velocity?

the density of the air
the height from which the body falls
the mass of the body
the shape of the body