Moments, levers & gears

A uniform wooden rod of mass 0.55 kg and length 1.3 m is attached to a wall by a light frictionless hinge.

A horizontal force F acts so that the rod hangs at an angle of 60.0° to the vertical.

What is the magnitude of F ?

The weight of a uniform bar is W. An object also of weight W is attached to one end. The bar is pivoted at the other end and held horizontal by a rope attached to its centre. The tension in the rope is 4 W.

What is angle θ ?

Figure 11 shows the stress–strain graph for a metal in tension up to the point at which it fractures.

Determine, using Figure 11, the Young modulus of the metal.

Explain how the graph shows that this metal is brittle.

Figure 12 shows a uniform rigid lighting beam AB suspended from a fixed horizontal support by two identical vertical steel wires. A lamp is attached to the midpoint of AB

The unloaded length of each steel wire was 1.20 m before it was attached to AB. AB is horizontal.

mass of AB = 4.4 kg mass of lamp = 16.0 kg distance between wires = 2.00 m diameter of each wire = 0.800 mm Young modulus of steel = 2.10 × 10¹¹ Pa

Calculate the extension of each wire.

The right-hand steel wire is removed and replaced with an aluminium wire of diameter 1.60 mm. The unloaded length of the aluminium wire is the same as that of the original steel wire.

When the lamp is at the midpoint of AB, one of the wires extends more than the other so that AB is not horizontal. To make AB horizontal the lamp has to be moved to a distance x from A. Figure 13 shows the new arrangement.

The Young modulus of aluminium is 7.00 × 10¹⁰ Pa.

Deduce distance x.

A heavy uniform trapdoor is hinged to the floor. It is held open by a rope as shown.

Which arrow shows the direction of the reaction force of the hinge on the trapdoor?

Figure 3 shows a garden gate with a pulley system designed to close the gate.

The pulley system raises weight A when the gate is opened. When the gate is released, A falls. The horizontal cable C passes over pulley R. The tension in cable C causes the gate to close.

Weight A is a solid cylinder with the following properties:

length = 0.23 m weight = 35 N

Table 2 gives the density of three available materials.

Deduce which one of the three materials is used for A.

Figure 4 shows the pulley arrangement when the gate is closed.

Pulleys P and M are frictionless so that the tension in the rope attached to A is equal to the weight of A. A weighs 35 N and the weight of moveable pulley M is negligible.

Calculate the tension in the horizontal cable C when the gate is closed.

Pulley M is pulled to the left as the gate is opened.

Explain why this increases the tension in the horizontal cable C.

Figure 5 shows a plan view with the gate open. The horizontal cable C passes over pulley R and is attached to the door at D. The angle between the door and the horizontal cable C is 12°.

The horizontal distance between the hinge and D is 0.95 m.

The tension in the horizontal cable C is now 41 N.

Calculate the moment of the tension about the hinge.

The same system is attached to an identical gate with stiffer hinges. Now the system does not supply a sufficiently large moment to close the gate.

Discuss two independent changes to the design to increase the moment about the hinges due to horizontal cable C.

A uniform plank weighs 1200 N and rests on two pillars P and Q. P and Q are 1.80 m apart. P is 0.20 m from one end of the plank. Q is 1.40 m from the other end of the plank. A man weighs 800 N and walks along the plank away from pillar P.

What is the horizontal distance between pillar P and the centre of mass of the man when the plank starts to tip?

A uniform piece of card in the shape of the letter L is suspended freely from a horizontal pin. A plumb line is also suspended from the pin. The diagram shows the card in its equilibrium position.

What is the position of the centre of mass of the piece of card?

The diagram shows the forces acting on a uniform rod.

Which statement is correct?

Figure 5 shows a simplified catapult used to hurl projectiles a long way.

The counterweight is a wooden box full of stones attached to one end of the beam. The projectile, usually a large rock, is in a sling hanging vertically from the other end of the beam. The weight of the sling is negligible. The beam is held horizontal by a rope attached to the frame.

The catapult is designed so that the weight of the beam and the weight of the empty wooden box have no effect on the tension in the rope.

Suggest how the pivot position achieves this.

The stones in the counterweight have a total mass of 610 kg and the projectile weighs 250 N.

Calculate the tension in the rope.

When the rope is cut, the counterweight rotates clockwise. When the beam is vertical it is prevented from rotating further. The projectile is then released horizontally with a velocity of 18 m s^–1, as shown in Figure 6.

The projectile is released at a height of 7.5 m above ground level.

The range of the catapult is the horizontal distance between the point where the projectile is released to the point where it lands.

Calculate the range. Ignore air resistance.

In another release, the sling is adjusted so that a projectile of the same mass is released just before the wooden beam is vertical. The projectile is not released horizontally.

Discuss the effect this change has on the range of the catapult.

Which statement about a couple is not true?

A non-uniform sign is 0.80 m long and has a weight of 18 N It is suspended from two vertical springs P and Q. The springs obey Hooke’s law and the spring constant of each spring is 240 N m⁻¹

The top end of spring P is fixed and the top end of spring Q is adjusted until the sign is horizontal and in equilibrium.

What is the extension of spring Q ?

This question is about using a digital balance to investigate the force on a wire placed in a magnetic field when there is an electric current in the wire.

A student carries out the procedure shown in Figure 1 and Figure 2. A metre ruler is pivoted at the 1.0 cm mark and a prism is placed on a digital balance. The free end of the ruler is raised and the balance is turned on and then set to zero, as shown in Figure 1.

The ruler is then supported by the prism with the apex of the prism at the 30.0 cm mark as shown in Figure 2. The height of the pivot is adjusted so that the ruler is horizontal.

Deduce the mass of the ruler. State one assumption you make.

The student attaches a uniform wire to the upper edge of the ruler, as shown in Figure 3.

The ends of the wire are connected to terminal blocks P and Q which are fixed firmly to the bench. A power supply and an ammeter are connected between P and Q.

These modifications cause the balance reading to increase slightly.

A horizontal uniform magnetic field is applied, perpendicular to the wire, between the 85 cm and 90 cm marks, as shown in the close-up diagram in Figure 3.

The balance reading M is recorded for increasing values of current I. A graph of these data is shown in Figure 4.

State and explain the direction of the horizontal uniform magnetic field.

It can be shown that B, the magnitude of the magnetic flux density of the horizontal uniform magnetic field, is given by

where change in force acting on the prism per unit current in the wire L = length of the region where the magnetic field cuts through the wire.

Determine B.

The experiment is repeated with the ruler pivoted at the 99.0 cm mark. Nothing else is changed from Figure 3.

This arrangement is shown in Figure 5.

Tick () one box in row 1 and one box in row 2 of Table 1 to identify the effect, if any, on the magnitude of the forces acting on the apparatus as a certain current is passed through the wire.

Tick () one box in row 3 and one box in row 4 of Table 1 to identify the effect, if any, on the graph produced for this modified experiment compared with the graph in Figure 4.

Figure 6 shows the balance being used to measure the forces between two wires. The connections joining these wires to the power supply are not shown.

The pan of the balance moves a negligible amount during use and it supports a straight conducting wire X of horizontal length L. Terminal blocks are used to connect X into the circuit. The weight of these does not affect the balance reading. A second conducting wire Y is firmly supported a distance d above X.

Show, by adding detail to Figure 6, the wire connections that complete the circuit. The currents in X and Y must have the same magnitude and be in the directions indicated.

The vertical force F on wire X due to the magnetic field produced by wire Y is given by

where k is a constant d is the perpendicular distance between X and Y I is the current in the wires and L is the horizontal length of wire X.

A student wants to measure k using the arrangement in Figure 6.

The student is told that the following restrictions must apply:

• L is fixed

• I must not exceed 5.0 A

• the result for k must be obtained using a graphical method

• the experimental procedure must involve only one independent variable.

Explain what the student could do to find k.

Figure 7

Explain how Figure 7 shows that the friction between the ladder and the wall is negligible.

The forces acting on the ladder are in equilibrium.

Draw an arrow on Figure 7 to show the direction of the resultant force from the ground acting on the ladder. Label your arrow G.

The ladder is 8.0 m long and weighs 390 N.

Calculate the magnitude of the resultant force from the wall on the ladder.

Suggest the changes to the forces acting on the ladder that occur when someone climbs the ladder .

Which diagram shows the bar in equilibrium?