# 2009 H2 Physics Tutorial: Forces

**S 2009 H2 PHYSICS TUTORIAL: FORCES**

## Hooke's Law

1.A weight is suspended from a spring. The variation with weight of the length of the spring is shown below.

What is the value of the spring constant (force constant) of the spring in N m-1?

*The graph only shows the length of the spring vs the weight of the load.*

*To find spring constant, the force acting on the spring and the extension need to be found.*

*Weight of the load provides the tension in the spring to extend it.*

Since F = kx

N m-1

2.The scale of a certain spring balance reads from 0 to 100 N and the scale of the spring is 0.25 m long. Find the elastic potential energy stored in the spring when the balance reads 20 N. (Assume that the spring obeys Hooke’s law.)

For a extension of 0.25 m, a force of 100 N is required.

Thus the spring constant can be obtained.

Since F = kx

N m-1

Elastic potential energy = ½ kx2

*with a force of 20 N, the extension will be (*=) 0.050 m

*Therefore elastic potential energy = ½ (400)(0.050)2 =0.50 J*

## Upthrust

3.A long narrow tube is filled with water of density 1020 kg m-3 to a depth of 1.00 m. The tube is then inclined at 30˚ to the horizontal as shown. If atmospheric pressure is 100 kPa, what is the pressure at point X, inside the tube?

*Point X is subjected to the force of the weight above it and the force due to the atmospheric pressure.*

*i.e. Px = Patm + ρfghfwhere hf is the vertical height of the fluid*

*= 100 × 103 + (1020 × 9.81 × 1.0 sin 30°)*

= 1.05 × 105 Pa

6. Fuel tanks often have gauges at the side which show how full the tank is (see Fig.6.1).

Explain why the pressure at the bottom of the gauge is the same as the pressure on the bottom of the tank.

*Pressure at a point in a fuel is due to the force acting on it. In this case, the force is the weight of the fluid above it. Even though the weight of the fluid in the tank is greater than the weight of the fluid in the gauge, the pressure at the bottom of both side is the same. This is because pressure is the ratio of the force over the area. The ratio of the weight over the area is the same for the tank and the gauge.*

4.An object hangs from a spring balance. The balance registers 50 N in air, 30 N when this object is immersed in water, and 44 N when the object is immersed in another liquid of unknown density. What is the density of the liquid?

In air / In water / In Liquid of unknown densityT1 = W

W = 50 N / T2 + U2 = W

U2 = 50 – 30 = 20 N / T3 + U3 = W

U3 = 50 – 44 = 6.0 N

Since U = ρfgVf

*For submerged in water, ρh20gVf = 20--- (1)*

*For submerged in unknown liquid, ρxgVf = 6.0--- (2)*

*Since the object is totally submerged in both liquid, the Vf is the same.*

(2) / (1) =

ρx = 300 kg m-3

5.A solid has a density of 3.5 g cm-3. Calculate the density of a liquid in which the solid would float with one-quarter of its total volume exposed above the liquid surface.

*When an object floats, it is stationary (i.e. no net force acting on it)*

*Weight of the object = Upthrust acting on the object*

mg = fgVf

ogVo= fgVf

*Since the volume of the object submerged is equal to the volume of the fluid displaced, *

Vf = ¾ Vo

3.5 × g × Vo= f × g × ¾ Vo

ρf = 4.7 g cm-3

## Equilibrium of coplanar forces

7.A metal disc is acted upon by a number of forces. The forces are all in the plane of the disc and the weight of the disc is negligible. In which of the following situations is the disc in equilibrium?

8.A bird of weight W lands at the midpoint of a horizontal wire stretched between two poles. The magnitude of the force exerted by each pole on the wire is F.

For equilibrium, Fn = 0

*∑Fx = 0:F cos θ = F cos θ (no use to solve this ques)*

∑Fy = 0:F sin θ + F sin θ = W

2 F sin θ = W

Since F > F sin θ

2F > 2 F sin θ

2F > W(Ans)

9.A small electrically charged sphere is suspended vertically from a thread. An oppositely charged rod is brought close to the sphere such that the sphere is in equilibrium when displaced from the vertical by an angle of 45°.

Which one of the following best represents the free body diagram for the sphere?

*For equilibrium, the net force and net torque acting on the object must be zero.*

i.e. Fn = 0 and τn = 0

*Thus if the vector addition of the forces is equal to 0, i.e. the force vectors forms a closed loop, the object is in equilibrium.*

- N98/2/1

Currents in the sea water cause the buoy to be displaced so that the rope makes an angle of 350 with the vertical, as shown in the above diagram. The buoy may be considered to be acted upon by 3 forces, the tension T in the rope, a horizontal force D and a vertical force V equal to 600 N.

(a)Using a scale of 1.0 cm to represent 100 N, complete the figure provided to produce a vector triangle for the three forces acting on the buoy.

(b) Use your vector triangle to determine the magnitude of the force D.

(a)

(b)By scale drawing: D = 410 N

By calculation:

D = 600 tan 35 = 420 N

11.The diagram shows a beam of length 20.0 m and mass 40.0 kg resting on two supports placed at 5.0 m from each end.

A girl of mass 50.0 kg stands on the beam between the supports. The reaction forces at the supports are shown.

(a)State the value of N1 + N2.

(b)The girl now moves towards end X of the beam. Determine the distance of the girl from end X when the beam is about to tip.

For equilibrium, Fn = 0 and τn = 0:

(a)Since Fn = 0 ∑Fy = 0

↑N1 + N2 = ↓ 50 × 9.81 + 40 × 9.81

N1 + N2 = 883 N

*(b)When the beam is about to tip, the contact at N1 will cease to occur, i.e. N1 = 0 *

Taking moment at N2,

40 × 9.81 × 5 = 50 × 9.81 × (5 – x)

x = 1 m

12.Rachid is using a wheelbarrow to move some blocks. When a lifting force is applied at the handle, its support legs are lifted off the ground. The dimensions of the wheelbarrow are shown in the diagram below.

When loaded, the total weight of the wheelbarrow and the blocks is 600 N. The ground is horizontal.

(a)Determine

(i)the minimum vertical force needed to lift the support legs off the ground.

(ii)the magnitude and the direction of the force exerted by the ground on the wheel.

Rachid now pushes the wheelbarrow forward **at constant speed**. He applies a force of 260 N to the handles at an angle of 50° to the vertical.

(b)(i)Calculate the horizontal component of the force exerted by Rachid.

(ii)Determine the magnitude of the resultant frictional force acting on the wheelbarrow.

*(a)(i)When the support legs are lifted from the ground, Ns = 0*

*Using principle of moments with pivot at the contact of the wheel and the ground,*

FL× 1.2 = 600 × 0.4

FL = 200 N;

(ii)Since Fn = 0

↑ FL + Nw = ↓ W

Nw = 600 – 200

= 400 N ↑

*(b)Since the wheelbarrow is moving at a constant speed in a straight line, Fn = 0.*

(i)Horizontal force = F sin 50°

= 260 sin 50°

= 199 N

(ii)∑Fx = 0

f = Fhorizontal = 199 N

13.A rod of length 1.0 metre has non-uniform composition, so that the centre of gravity is not at its geometrical centre. The rod is laid on supports across two top-pan balances as shown in the diagram. The balances (previously set at zero) give readings of 240g and 320g.

*Since the rod is stationary, it is in equilibrium, i.e. Fn = 0 andτn = 0*

*Taking moments about C.G (so that the moment due to the W of the rod is zero)*

240 L = 320(1 – L)

L = 0.57 m

*Therefore, c.g. of rod is 0.070 m to the right of the geometrical center.*

14.A mountain climber is rappelling down a vertical wall (the figure below ). The rope attaches to a buckle strapped to the climber's waist 15.0 cm to the right of his center of gravity. If the climber weighs 770 N, find

(a)the tension in the rope and

(b)the magnitude and direction of the contact force exerted by the wall on the climber's feet.

*Since the person is not moving, he is in equilibrium, i.e. Fn = 0 and τn = 0*

*(a)The pivot chosen is the point of contact of the wall with the climber’s feet (this is so that the moment due to the friction and the normal force due to the wall will be 0.)*

Using ∑MA = 0,

Sum of anti-clockwise moment = Sum of clockwise moment

T cos25˚ x 106 = 770 x 91

T = 729 N

(b)∑Fx = 0

N = T sin θ

N = 729 sin 25°

= 308 N

∑Fy = 0

f + T sin θ = Wc

f = 770 – 729 cos 25°

= 109 N

R2 =3082 + 1092

R = 327 N

θ = 19.5°

R is 327 N above the horizontal.

15A ladder of length 5.0 m and mass 2.0 kg rests against a wall. Its upper end is at a height of 4.0 m above the ground. Assuming that the wall is smooth, but not the ground, what is the force exerted by the wall on the ladder?

Since the ladder is at rest, it is in equilibrium,

i.e. Fn = 0 and τn = 0

Take moments about bottom of ladder

(so that the moment due to X and Y is zero):

mg x 1.5 = N x 4.0

N = (2.0 x 9.81) x 1.5 ÷ 4.0 = 7.36 N

Additional:

∑Fx = 0 → X = N = 7.36 N

∑Fy = 0 → Y = mg = 2.0 x 9.81 = 19.6 N