Give the sign of work done: by a person lifting a bucket, by gravity on that bucket, by friction on a sliding body, by an applied force maintaining uniform rough-surface motion, and by air resistance on a pendulum.
Solution to Question 21
Complete workingThe sign follows from W=Fs cosθ: positive for force along displacement and negative for force opposite displacement.
State the net force on: a raindrop falling at constant speed, a cork floating at rest, a stationary kite, a car moving at constant velocity on a rough road, and an electron far from matter and fields.
Solution to Question 22
Complete workingIn every case the acceleration is zero. Newton's second law therefore gives the same conclusion even though individual forces may be present.
Classify volume, mass, speed, acceleration, density, amount of substance, velocity, angular frequency, displacement and angular velocity as scalars or vectors.
Solution to Question 23
Complete workingA scalar needs magnitude only; a vector requires magnitude and direction and follows vector addition.
Locate the centre of mass of a uniform sphere, cylinder, ring and cube. Must a centre of mass always lie inside the material?
Solution to Question 24
Complete workingEvery symmetry operation that leaves a uniform body unchanged must also leave its centre of mass unchanged. Their symmetry axes therefore intersect at the centre.
A die is rolled. Let E be '4 appears' and F be 'an even number appears'. Decide whether E and F are mutually exclusive.
Solution to Question 25
Complete workingDefine the sample space and count each favourable outcome once.
The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.
For 4,7,8,9,10,12,13,17, find the mean deviation about the mean.
Solution to Question 26
Complete workingForm the required totals explicitly and divide by the full number of observations.
The variance is nonnegative and the reported standard deviation is its positive square root.
Evaluate lim(x→3)(x+3).
Solution to Question 27
Complete workingSimplify the expression before substitution and preserve one-sided information.
Substitution into the original expression verifies the stated domain and final value.
A point lies on the x-axis. State its y- and z-coordinates.
Solution to Question 28
Complete workingUse rectangular-coordinate definitions and keep corresponding coordinates aligned.
The final relation satisfies the original coordinate or distance condition, completing the verification.
Find the circle with centre (0,2) and radius 2.
Solution to Question 29
Complete workingTranslate every condition into an equation and retain exact values throughout the conic calculation.
The last line satisfies the original conditions and supplies every requested part.
Plot A(−4,5), B(0,7), C(5,−5), and D(−4,−2) in that order, then calculate the area of quadrilateral ABCD.
Solution to Question 30
Complete workingTranslate every condition into an equation and retain exact values throughout the straight-line calculation.
The last line satisfies the original conditions and supplies every requested part.