In this chapter, you will learn
- —Understand how time has been measured from ancient to modern devices
- —Learn the concept of a simple pendulum and its time period
- —Define speed and calculate it using the formula Speed = Distance / Time
- —Differentiate between uniform and non-uniform motion
- —Read and interpret distance-time graphs
- —Convert units of speed between m/s and km/h
- —Understand the difference between speedometer and odometer
Measurement of Time
Humans have always needed to measure time. Over the centuries, many devices have been developed for this purpose:
1. Sundial: One of the oldest devices. It uses the shadow of a vertical stick (gnomon) cast by the Sun to tell the time. As the Sun moves across the sky, the shadow moves, indicating the time. It cannot work at night or on cloudy days.
2. Water Clock: Time is measured by the flow of water from one container to another through a small hole. The level of water indicates the time elapsed.
3. Sand Clock (Hourglass): Sand flows from an upper bulb to a lower bulb through a narrow neck. It measures a fixed interval of time.
4. Pendulum Clock: Uses the regular back-and-forth motion of a pendulum to keep time. Invented by Christiaan Huygens in 1656.
5. Quartz Clock: Uses vibrations of a quartz crystal to measure time very accurately. Most modern clocks and wristwatches use quartz crystals.
6. Atomic Clock: The most accurate time-measuring device. It uses vibrations of cesium atoms. It may lose or gain only one second in millions of years.
The SI unit of time is the second (s). Other units include minutes (min), hours (h), days, and years.
Exam Tip
Remember the order of time-measuring devices from least to most accurate: Sundial, Water clock, Sand clock, Pendulum clock, Quartz clock, Atomic clock.
Simple Pendulum
A simple pendulum consists of a small heavy metal ball (called a bob) suspended from a rigid support by a light, inextensible string.
When the bob is pulled to one side and released, it swings back and forth. This back-and-forth motion is called oscillatory motion or periodic motion.
Key Terms:
- Mean position (Rest position): The position of the bob when it is at rest (hanging straight down).
- Extreme position: The farthest point the bob reaches on either side of the mean position.
- One oscillation: One complete to-and-fro movement of the bob. For example, from one extreme to the other extreme and back again, or from the mean position to one extreme, to the other extreme, and back to the mean position.
- Amplitude: The maximum displacement of the bob from its mean position. It is the distance from the mean position to an extreme position.
Exam Tip
One oscillation is one complete to-and-fro motion. If the bob goes from A to B and back to A, that is one oscillation. Going only from A to B is half an oscillation.
Common Mistake
Students often count one swing (from one side to the other) as one full oscillation. Remember: one complete oscillation means the bob returns to its starting position moving in the same direction.
Time Period of a Pendulum
The time period (T) of a pendulum is the time taken to complete one full oscillation.
Time Period (T) = Total time taken / Number of oscillations
Example: If a pendulum completes 20 oscillations in 40 seconds, then:
T = 40 s / 20 = 2 seconds
This means the pendulum takes 2 seconds for each complete oscillation.
Key Fact: The time period of a simple pendulum depends only on its length. It does NOT depend on the weight (mass) of the bob or the amplitude of the swing (for small amplitudes).
- Longer pendulum = Longer time period (swings slower)
- Shorter pendulum = Shorter time period (swings faster)
To measure the time period accurately, we count many oscillations (say 20) and divide the total time by the number of oscillations.
Exam Tip
In numerical problems, always calculate time period by dividing total time by number of oscillations. Never try to time a single oscillation directly as it introduces errors.
Frequency of a Pendulum
The frequency of a pendulum is the number of oscillations it completes in one second.
Frequency (f) = Number of oscillations / Total time = 1 / Time Period
The SI unit of frequency is hertz (Hz). 1 Hz means 1 oscillation per second.
Relationship between Time Period and Frequency:
f = 1 / T and T = 1 / f
Example: If the time period of a pendulum is 0.5 seconds, its frequency is:
f = 1 / 0.5 = 2 Hz
This means the pendulum completes 2 oscillations every second.
Exam Tip
Frequency and time period are inversely related. If T increases, f decreases and vice versa. This is a very common numerical question.
Speed
Speed is the distance covered by an object in a given time. It tells us how fast or slow an object is moving.
Speed = Distance / Time
The SI unit of speed is metre per second (m/s). Another commonly used unit is kilometre per hour (km/h).
Example 1: A car travels 150 km in 3 hours. What is its speed?
Speed = Distance / Time = 150 km / 3 h = 50 km/h
Example 2: A boy runs 100 metres in 20 seconds. What is his speed?
Speed = Distance / Time = 100 m / 20 s = 5 m/s
We can also rearrange the formula:
- Distance = Speed x Time
- Time = Distance / Speed
Exam Tip
Always check the units in the question. If distance is in km and time in hours, speed will be in km/h. If distance is in metres and time in seconds, speed will be in m/s.
Common Mistake
Students sometimes mix units, for example using kilometres with seconds. Always ensure distance and time are in matching units before calculating speed.
Units of Speed and Conversions
The two most commonly used units of speed are:
- m/s (metres per second) - SI unit
- km/h (kilometres per hour) - commonly used for vehicles
Conversion between km/h and m/s:
To convert km/h to m/s: multiply by 5/18
To convert m/s to km/h: multiply by 18/5
Why 5/18?
1 km/h = 1000 m / 3600 s = 5/18 m/s
Example 1: Convert 72 km/h to m/s:
72 x 5/18 = 20 m/s
Example 2: Convert 15 m/s to km/h:
15 x 18/5 = 54 km/h
Exam Tip
The conversion factor 5/18 is very important. Quick check: 36 km/h = 10 m/s. Use this to verify your answers.
Speedometer and Odometer
Vehicles have two important instruments for measuring motion:
Speedometer: An instrument that shows the instantaneous speed of a vehicle at any moment. It tells us how fast the vehicle is moving right now. The reading on a speedometer keeps changing as the vehicle speeds up or slows down.
Odometer: An instrument that measures the total distance travelled by the vehicle. Unlike the speedometer, the odometer reading only increases - it never decreases. It records the cumulative distance from the time the vehicle was manufactured.
Key Difference:
- Speedometer measures speed (how fast) at a given instant
- Odometer measures distance (how far) travelled in total
Exam Tip
Remember: Speedometer = Speed (instantaneous), Odometer = Distance (total). Odo- comes from the Greek word 'hodos' meaning path or way.
Uniform Motion
An object is said to be in uniform motion when it covers equal distances in equal intervals of time.
Example: A car travelling on a straight highway at a constant speed of 60 km/h. In every hour, it covers exactly 60 km.
| Time (hours) | Distance (km) |
|---|---|
| 1 | 60 |
| 2 | 120 |
| 3 | 180 |
| 4 | 240 |
In uniform motion, the speed remains constant throughout the journey. The distance-time graph for uniform motion is a straight line.
Note: Uniform motion in a straight line is very rare in daily life. Most real-world motion is non-uniform.
Exam Tip
Uniform motion means CONSTANT speed. The distance-time graph is always a straight line for uniform motion.
Non-Uniform Motion
An object is said to be in non-uniform motion when it covers unequal distances in equal intervals of time.
Example: A car moving in city traffic. It speeds up, slows down, and stops at traffic signals. The distance covered each minute is different.
| Time (minutes) | Distance covered (m) |
|---|---|
| 1 | 200 |
| 2 | 350 |
| 3 | 500 |
| 4 | 580 |
In non-uniform motion, the speed keeps changing. The distance-time graph for non-uniform motion is a curved line.
For non-uniform motion, we use the concept of average speed:
Average Speed = Total distance travelled / Total time taken
Example: A bus travels 100 km in the first 2 hours and 60 km in the next 2 hours.
Total distance = 100 + 60 = 160 km
Total time = 2 + 2 = 4 hours
Average speed = 160 / 4 = 40 km/h
Exam Tip
Average speed is NOT the average of two speeds. Always use Total Distance / Total Time. This is one of the most common errors in exams.
Common Mistake
Students often calculate average speed as (speed1 + speed2) / 2. This is WRONG. Average speed = Total distance / Total time.
Distance-Time Graph
A distance-time graph shows how the distance covered by an object changes with time. Time is plotted on the X-axis (horizontal) and distance is plotted on the Y-axis (vertical).
How to read a distance-time graph:
- Straight line (going upward): The object is in uniform motion (constant speed). A steeper line means higher speed.
- Curved line (going upward): The object is in non-uniform motion (changing speed). If the curve bends upward, speed is increasing. If it bends downward, speed is decreasing.
- Horizontal line (flat): The object is stationary (not moving). Distance is not changing even though time is passing.
Calculating speed from the graph:
The speed of the object equals the slope of the distance-time graph.
Speed = (Distance at point 2 - Distance at point 1) / (Time at point 2 - Time at point 1)
Example: If at time = 2 s the distance is 10 m and at time = 5 s the distance is 25 m:
Speed = (25 - 10) / (5 - 2) = 15 / 3 = 5 m/s
Exam Tip
Always label the axes correctly: X-axis = Time, Y-axis = Distance. The slope of the graph gives the speed. A steeper slope means faster speed.
Common Mistake
Students sometimes swap the axes, putting distance on the X-axis and time on the Y-axis. Always remember: Time goes on X-axis, Distance on Y-axis.
Numerical Problem Solving - Speed, Distance, and Time
Let us solve some typical numerical problems using the three formulas:
Speed = Distance / Time
Distance = Speed x Time
Time = Distance / Speed
Problem 1: A train travels 360 km in 4 hours. Find its speed in (a) km/h and (b) m/s.
(a) Speed = 360 km / 4 h = 90 km/h
(b) Speed in m/s = 90 x 5/18 = 25 m/s
Problem 2: A cyclist moves at 10 m/s. How far will he travel in 5 minutes?
Time = 5 min = 5 x 60 = 300 s
Distance = Speed x Time = 10 x 300 = 3000 m = 3 km
Problem 3: How long will a car take to cover 450 km at a speed of 75 km/h?
Time = Distance / Speed = 450 / 75 = 6 hours
Exam Tip
Always convert time to the correct unit before solving. If speed is in m/s, convert minutes to seconds. If speed is in km/h, convert minutes to hours.
Numerical Problem Solving - Pendulum
Let us practise pendulum calculations:
Problem 1: A pendulum completes 40 oscillations in 80 seconds. Find its time period and frequency.
Time period (T) = Total time / Number of oscillations
T = 80 / 40 = 2 seconds
Frequency (f) = 1 / T = 1 / 2 = 0.5 Hz
Problem 2: The frequency of a pendulum is 4 Hz. How many oscillations does it complete in 1 minute?
Frequency = 4 Hz means 4 oscillations per second
In 1 minute (60 seconds):
Number of oscillations = 4 x 60 = 240 oscillations
Problem 3: A pendulum has a time period of 0.4 seconds. What is its frequency? How many oscillations will it make in 2 minutes?
Frequency = 1 / T = 1 / 0.4 = 2.5 Hz
In 2 minutes = 120 seconds
Number of oscillations = 2.5 x 120 = 300 oscillations
Exam Tip
Pendulum numericals are very common in exams. Always write the formula first, then substitute values. Show all steps clearly.