A block of mass 1.5 kg is at rest on a rough surface which is inclined at 20° to the horizontal as shown.

a. Draw a free-body diagram showing the three forces acting on the block.
b. Calculate the component of the weight that acts down the slope.
c. Use your answer to part b to determine the force of friction that acts on the block.
d. If the angle of the surface is actually measured as 19° and 21° determine the absolute uncertainty in this angle and the uncertainty this produces in the value for part b.
e. Determine the normal contact force between the block and the surface.

a. State what is meant by:
i. A couple
ii. Torque.
b. The engine of a car produces a torque of 200 N m on the axle of the wheel in contact with the road. The car travels at a constant velocity towards the right:

i. Copy the diagram of the wheel and show the direction of rotation of the wheel, and the horizontal component of the force that the road exerts on the wheel.
ii. State the resultant torque on the wheel. Explain your answer.
iii. The diameter of the car wheel is 0.58 m. Determine the value of the horizontal component of the force of the road on the wheel.

This diagram shows a picture hanging symmetrically by two cords from a nail fixed to a wall. The picture is in equilibrium.

a. Explain what is meant by equilibrium.
b. Draw a vector diagram to represent the three forces acting on the picture in the vertical plane. Label each force clearly with its name and show the direction of each force with an arrow.
c. The tension in the cord is 45 N and the angle that each end of the cord makes with the horizontal is 50°. Calculate:
i. The vertical component of the tension in the cord
ii. The weight of the picture.

a. State the two conditions necessary for an object to be in equilibrium.
b. A metal rod of length 90 cm has a disc of radius 24 cm fixed rigidly at its centre, as shown. The assembly is pivoted at its centre.

Two forces, each of magnitude 30 N, are applied normal to the rod at each end so as to produce a turning effect on the rod. A rope is attached to the edge of the disc to prevent rotation.
Calculate:
i. The torque of the couple produced by the 30 N forces
ii. The tension T in the rope.

This is a velocity–time graph for a vertically bouncing ball.

The ball is released at A and strikes the ground at B. The ball leaves the ground at D and reaches its maximum height at E. The effects of air resistance can be neglected.
a. State:
i. Why the velocity at D is negative
ii. Why the gradient of the line AB is the same as the gradient of line DE
iii. What is represented by the area between the line AB and the time axis
iv. Why the area of triangle ABC is greater than the area of triangle CDE.

b. The ball is dropped from rest from an initial height of 1.2 m. After hitting the ground the ball rebounds to a height of 0.80 m. The ball is in contact with the ground between B and D for a time of 0.16 s.
Using the acceleration of free fall, calculate:
i. The speed of the ball immediately before hitting the ground
ii. The speed of the ball immediately after hitting the ground
iii. The acceleration of the ball while it is in contact with the ground. State the direction of this acceleration.

A car driver is travelling at speed v on a straight road. He comes over the top of a hill to find a fallen tree on the road ahead. He immediately brakes hard but travels a distance of 60 m at speed v before the brakes are applied. The skid marks left on the road by the wheels of the car are of length 140 m, as shown.

The police investigate whether the driver was speeding and establish that the car decelerates at 2.0 m s⁻² during the skid.

a. Determine the initial speed v of the car before the brakes are applied.

b. Determine the time taken between the driver coming over the top of the hill and applying the brakes. Suggest whether this shows whether the driver was alert to the danger.

c. The speed limit on the road is 100 km/h. Determine whether the driver was breaking the speed limit.

A student measures the speed v of a trolley as it moves down a slope. The variation of v with time t is shown in this graph.

a. Use the graph to find the acceleration of the trolley when t = 0.70 s.
b. State how the acceleration of the trolley varies between t = 0 and t = 1.0 s.
Explain your answer by reference to the graph.
c. Determine the distance travelled by the trolley between t = 0.60 and t = 0.80 s.
d. The student obtained the readings for v using a motion sensor. The readings may have random errors and systematic errors. Explain how these two types of error affect the velocity–time graph.

A hot-air balloon rises vertically. At time t = 0, a ball is released from the balloon. This graph shows the variation of the ball’s velocity v with t. The ball hits the ground at   t = 4.1 s.

a. Explain how the graph shows that the acceleration of the ball is constant.
b. Use the graph to:
i. Determine the time at which the ball reaches its highest point
ii. Show that the ball rises for a further 12 m between release and its highest point
iii. Determine the distance between the highest point reached by the ball and the ground.
c. The equation relating v and t is v = 15 − 9.81t. State the significance in the equation of:
i. The number 15
ii. The negative sign.

If you drop a stone from the edge of a cliff, its initial velocity u = 0, and it falls with acceleration g = 9.81 m s⁻². You can calculate the distance s it falls in a given time t using an equation of motion.
a. Copy and complete Table 2.3, which shows how s depends on t.
b. Draw a graph of s against t.
c. Use your graph to find the distance fallen by the stone in 2.5 s.
d. Use your graph to find how long it will take the stone to fall to the bottom of a cliff 40 m high.
Check your answer using the equations of motion.

A steel ball falls from rest through a height of 2.10 m. An electronic timer records a time of 0.67 s for the fall.
a. Calculate the average acceleration of the ball as it falls.
b. Suggest reasons why the answer is not exactly 9.81 m s⁻².
c. Suppose the height is measured accurately but the time is measured to an uncertainty of ±0.02 s.

Calculate the percentage uncertainty in the time and the percentage uncertainty in the average acceleration. You can do this by repeating the calculation for g using a time of 0.65 s. You can find out more about uncertainty in Chapter P1.