Worked example 1
Worked example 2
Example: Vector R makes an angle θ with the x-axis. R is broken into component vectors Rx and Ry.
cos θ = adj = R x
hyp R
∴ Rcos θ = Rx
sin θ = opp = R y
hyp R
∴ Rsin θ = Ry
Rx = a cos θ and Ry = sin Ry, where Ry is the angle between a and the x-axis (Rx).
Worked example 3
If vector R has a magnitude of 5 and is at an angle of θ = 36,86°, the components are Fx = 5 cos 36,86° = +4 and Ry = 5 sin 36,86° = +3.
Worked example 4
Draw a force diagram and a free body diagram for an object hanging from a rope or a cable.
Solution |
Worked example 5
If an object with a mass of 40 kg slides down a surface which has a coefficient of kinetic friction μk = 0,14, and a slope of 15°, what is the net force on the object as it slides down the surface? Use the diagram to help you.
Solution Weight component down the slope =Fg sin θ = mg sin 15° = 40 × 9,8 × 0,26 = 101,92 N. Frictional force up the slope Fg·μk cos θ = 40 × 9,8 × 0,14 × 0,96 = 52,68 N ∴ Net force down the slope is: |
Worked example 6
Calculate the components of the weight of an object when it is resting on a surface which slopes at an angle of θ.
Solution θ + a = 90° and a + c = 90°. So c = θ. The component of the weight perpendicular to the surface with a slope of θ is Fg⊥ =Fg cos θ and parallel to the surface is Fg|| =Fg sin θ. |
Worked example 7
John exerts a force of 100 N to the right on a box resting on a rough, horizontal surface. Sarah exerts a force of 50 N to the left on the box. The friction between the box and the surface is 5 N. Draw a force diagram and calculate the resultant force acting on the box.
Solution Fnet = ΣF = (+FJohn) + (−FSarah) + (−Ff) =FJohn –FSarah –Ff = 100 – 50 – 5 = 45 N ∴Fnet= 45 N to the right |
Activity 1
A box of mass 100 kg slides down a rough slope which forms an angle of 30° to the horizontal. The friction that acts on the box is 20 N. Draw a free body diagram representing all the forces acting on the object and calculate the resultant force acting on the box and causing it to slide. Perpendicular forces may be ignored. (12)
Solution Let the direction down the slope be positive. Then Ff = –20N Fnet = ΣF = (+Fgll) + (Ff ) = Fgll + Ff = mg·sin α + Ff = (100)(9,8)(sin 30°)+( – 20) = 490 – 20 = 470 N ∴ 470 N, down the slope |
[12]
Activity 2
R and S are two positively charged spheres. P is a negatively charged sphere. Sphere R exerts an electrostatic force of 0,2 N on P and sphere S exerts a force of 0,6 N on sphere P.
Draw a free body diagram for sphere P and then calculate the resultant force on sphere P. (5)
Solution F net = ΣF = (FS on P) + (FR on P) = 0,6 + (–0,2) = 0,4 N ∴ 0,4 N towards S [5] Note:FR = –0,2 N |
Remember: Opposite charges attract. Therefore R attracts P and S attracts P.
Activity 3
Three identical spheres X, Y and Z are in the same horizontal plane.
Spheres X and Z are both positive and sphere Y is negative. Sphere Y exerts an electrostatic force of 450 N on sphere X and sphere Z exerts an electrostatic force of 350 N on sphere X.
Solutions
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Worked example 8
The diagram shows the forces on a trolley moving with constant velocity.
Solutions
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Worked example 9
A resultant force F is applied to an object of mass m and the object accelerates at a .
What will the object’s acceleration be if the resultant force acting on the object is tripled?
Solution m is constant ∴ a ∝ F and if the force is tripled (from F to 3F ), the acceleration will also triple ∴ the object will accelerate at 3a . |
Worked example 10
A constant resultant force
F is applied to objects of masses m and 2 m. If the object of mass m accelerates at a , what will the acceleration of the other object be?
Solution F is constant ∴ a ∝ 1/m If the mass doubles (from m to 2 m), the acceleration will halve ∴ the object of mass 2 m accelerates at ½ a . |
Activity 4
Two boxes, A and B, are lying on a table and are connected by a piece of string. The mass of box A is 3 kg and the mass of box B is 2 kg. Assume that the mass of the string is very small, so we can ignore it. A 30 N pulling force, pointing to the right, is applied to box B, causing the two boxes to move. The surface acts with a frictional force of 5,9 N on box A and 4,1 N on box B.
Solution
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Activity 5
The sketch below shows a block of 8,5 kg at equilibrium on an inclined (sloping) plane (surface).
Calculate:
Solution
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Activity 6
Two blocks of 25 kg and 15 kg are connected by a light string on a horizontal surface. Assume that the string cannot stretch. A force of magnitude 240 N is applied to the block of 15 kg forming an angle of 60° with the horizontal as shown in the sketch below. The coefficient of kinetic friction is 0,20.
Solutions
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Activity 7
The sketch below shows two blocks connected by a string of negligible mass that passes over a frictionless pulley also of negligible mass.
The arrangement is known as Atwood’s machine. One block has mass
m1 = 2 kg and the other has mass m2= 4 kg.
The blocks have just this instant been released from rest.
Solutions
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Activity 8
A 4 kg block on a horizontal, rough surface is connected to a 8 kg block by a light string that passes over a frictionless pulley as shown below. Assume that the string cannot stretch. The coefficient of kinetic (dynamic) friction between the block of 4 kg and the surface is 0,6.
Solutions
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Worked example 11
A force of gravitational attraction exists between the earth with mass m1 and a person with mass m2. The force on m1 is F1 and the force on m2 is F2.
Compare the magnitudes (sizes) of these forces and state the name of the law which explains your answer.
Solution
F1 = – F2 according to Newton’s Third Law of Motion:
The force between the earth (m1) and a person (m2) standing on its surface:
∴ m2g = G m 1m 2
r2
∴ g = G m1
r2
Worked example 12
The diagram shows a ball A of mass 0,01 kg which is 1 m (measured from centre to centre) from another ball B of mass 520 g. Calculate the magnitude of the force of ball A on ball B.
Solution
F = G m 1m 2 = 6.67 x 10-11 x 0.01 x 0.52
r2 12
= 3,57 × 10–14 N
It is important to understand and be able to compare mass and weight.
Worked example 13
An object weighs 720 N on earth. It orbits the Earth in a satellite at a height equal to the earth’s diameter, above the surface of the Earth. What does the object weigh on the satellite?
Hint: diameter = 2 × radius
Step by step
Step 1. Determine the number of radii from the centre of the Earth.
On the Earth’s surface, the object is 1 radius from the Earth’s centre.
Step 2. Determine how many times the distance between the object and the centre of the Earth has increased.
In orbit, the object is 1 diameter = 2 radii above the surface.
Step 3. Square this number (multiply it by itself).
So the object is 3 radii from the centre. (3)2 = 9
Step 4. The force has decreased this number of times because F ∝ 1
r2
∴ the gravitational force on the object has decreased 9 times because F ∝ 1
r2
Step 5. Divide the value of the force (or weight) by the value calculated in step 3.
∴ its weight on the satellite is 720 N ÷ 9 = 80 N
Gravitational acceleration on planets other than earth
Newton’s universal law of gravitation can be used to calculate the acceleration due to the force of gravity on any planet.
If the mass and radius of a planet are known, we can calculate g for that planet.
Worked example 14
The Mars Rover is an automated vehicle that has been sent to explore the surface of the planet Mars.
If the value of acceleration due to gravity on the planet Mars is g Mars = 3,7 m·s–2.
Calculate the weight of the Mars rover on Mars if it has a mass of 174 kg.
Solution
w Mars = g Mars × mobject = 3,7 × 174 = 643,8 N towards the centre of the planet Mars.