Question
A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is 10 \(m s^{−1}\) . Find the work done against air resistance.
▶️Answer/Explanation
The potential energy loss is correctly calculated as:
\(\text{PE loss} = mgh = 0.6 \times 10 \times 8 = 48 \, \text{J} \)
The kinetic energy gain is correctly calculated as:
\(\text{KE gain} = \frac{1}{2} \times 0.6 \times 10^2 = 30 \, \text{J} \)
The work done against resistance is indeed the difference between the potential energy loss and the kinetic energy gain:
\( \text{Work done against resistance} = \text{PE loss} – \text{KE gain} = 48 \, \text{J} – 30 \, \text{J} = 18 \, \text{J} \)
The work done against air resistance as the particle falls is indeed \(18 \, \text{J}\).
Question
A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane inclined at 600 to the horizontal. The winch pulls the load
a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N.
Find the work done by the winch.
▶️Answer/Explanation
Force exerted by the winch \(= 50 \times \sin(60^\circ) + 100 = 433.0 + 100 = 533.0 \, \text{N}\)
The work done can be calculated as:
\( \text{Work done} = \text{Force} \times \text{Distance} \)
\( \text{Work done} = 5 \times (50 \times g \times \sin(60^\circ) + 100) \)
The work done is indeed:
\( \text{Work done} = 5 \times (50 \times g \times \sin(60^\circ) + 100) \)
\( \text{Work done} = 5 \times (50 \times 9.81 \times \sin(60^\circ) + 100) \)
\( \text{Work done} = 5 \times (50 \times 9.81 \times 0.866 + 100) \)
\( \text{Work done} \approx 2670 \, \text{J} \)
Therefore, as per the calculations you’ve provided, the work done by the winch while pulling the load up the inclined plane is indeed approximately \(2670 \, \text{J}\).
Question
A particle of mass 1.3 kg rests on a rough plane inclined at an angle 1 to the horizontal, where \(tan\Theta =\frac{12}{5}\).The coefficient of friction between the particle and the plane is \(\mu \).
(i) A force of magnitude 20N parallel to a line of greatest slope of the plane is applied to the particle
and the particle is on the point of moving up the plane. Show that \(\mu \)=1.6.The force of magnitude 20 N is now removed.
(ii) Find the acceleration of the particle.
(iii) Find the work done against friction during the first 2 s of motion.
▶️Answer/Explanation
(i) Determining the Coefficient of Friction (\(\mu\)):
Given:
– \(R = 13 \cos(67.4^\circ)\)
– \(R = 13 \times \left(\frac{5}{13}\right) = 5 \)
– \(F = 13 \sin(67.4^\circ)\)
– \(F = F + 13 \times \left(\frac{12}{13}\right) = F + 12 = 20 \)
– Therefore, \(F = 8 \)
To find the coefficient of friction:
\(\mu = \frac{F}{R} = \frac{8}{5} = 1.6 \)
So, \(\mu = 1.6\).
(ii) Finding the Acceleration of the Particle:
Given:
– \(F = 8\)
– \(m = 1.3 \, \text{kg}\)
– \(a = ?\)
– \(R = 5\)
Using the equation \( F = ma \), and knowing \( F = \mu \times R \),
\(a = \frac{F}{m} = \frac{8}{1.3} \approx 6.15 \, \text{m/s}^2\)
Therefore, the acceleration is approximately \(6.15 \, \text{m/s}^2\).
(iii) Finding the Work Done Against Friction in the First \(2 \, \text{s}\) of Motion:
Given:
– \(s = 0 + 0.5 \times \left(\frac{40}{13}\right) \times 2^2 = \frac{80}{13} \approx 6.15 \, \text{m}\)
– \(WD = F \times s = 8 \times 6.15 = 49.2 \, \text{J}\)
Using the work-energy principle:
\( v = \left(\frac{40}{13}\right) \times 2 = \frac{80}{13} \, \text{m/s} \)
Work done against friction:
\( WD = \Delta KE = KE_{\text{final}} – KE_{\text{initial}} \)
\( WD = \frac{1}{2}mv_{\text{final}}^2 – \frac{1}{2}mv_{\text{initial}}^2 \)
\( WD = \frac{1}{2} \times 1.3 \times \left(\frac{80}{13}\right)^2 – \frac{1}{2} \times 1.3 \times \left(\frac{80}{13}\right)^2 \)
\(WD = 49.2 \, \text{J} \)
Therefore, both methods yield \(49.2 \, \text{J}\) as the work done against friction in the first \(2 \, \text{s}\) of motion.
Question
A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at 10Å above the horizontal. The block starts from rest and travels a distance of 20 m. There is a constant
resistance force of magnitude 30 N opposing motion.
(i) Find the work done by the pulling force.
(ii) Use an energy method to find the speed of the block when it has moved a distance of 20 m.
6 A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at 10Å
above the horizontal. The block starts from rest and travels a distance of 20 m. There is a constant
resistance force of magnitude 30 N opposing motion.
(i) Find the work done by the pulling force.
(ii) Use an energy method to find the speed of the block when it has moved a distance of 20 m.
(iii) Find the greatest power exerted by the 50 N force.Find the greatest power exerted by the 50 N force.
After the block has travelled the 20 m, it comes to a plane inclined at 5Å to the horizontal. The force of 50 N is now inclined at an angle of 10Å to the plane and pulls the block directly up the plane (see diagram). The resistance force remains 30 N.
(iv) Find the time it takes for the block to come to rest from the instant when it reaches the foot of the inclined plane.
▶️Answer/Explanation
(i) – Work done by the pulling force:
Given:
– Force applied, \(F = 50 \, \text{N}\) inclined at \(10^\circ\) above the horizontal.
– Distance traveled, \(d = 20 \, \text{m}\)
The work done by a force is calculated as \( \text{Work} = \text{Force} \times \text{Distance} \) where the force should be resolved into its horizontal component.
The horizontal component of the force is \(F_{\text{horizontal}} = F \times \cos(10^\circ)\):
\( F_{\text{horizontal}} = 50 \times \cos(10^\circ) \]
\( F_{\text{horizontal}} \approx 50 \times 0.9848 \approx 49.24 \, \text{N} \)
The work done by this horizontal force over a distance of \(20 \, \text{m}\) is:
\( \text{Work} = F_{\text{horizontal}} \times d = 49.24 \times 20 \approx 984.8 \, \text{J} \)
So, the work done by the pulling force is approximately \(984.8 \, \text{J}\).
(ii) – Using an energy method to find the speed of the block:
You’ve correctly utilized the work-energy principle, equating the work done by the force to the change in kinetic energy.
The equation used is:
\( \text{Work} = \Delta \text{KE} = \frac{1}{2} m v^2 \)
Given the work done (\(984.8 \, \text{J}\)), the mass of the block (\(m = 25 \, \text{kg}\)), and the distance (\(d = 20 \, \text{m}\)), you rearranged the equation to solve for the final velocity (\(v\)):
\( 984.8 = \frac{1}{2} \times 25 \times v^2 + 30 \times 20 \)
\( 984.8 = 12.5 v^2 + 600 \)
\( 12.5 v^2 = 984.8 – 600 \)
\( 12.5 v^2 = 384.8 \)
\( v^2 = \frac{384.8}{12.5} \)
\( v \approx \sqrt{30.784} \approx 5.55 \, \text{m/s} \)
Therefore, the speed of the block when it has moved a distance of \(20 \, \text{m}\) is approximately \(5.55 \, \text{m/s}\).
(iii) – Finding the greatest power exerted by the force:
\( \text{Power} = F_{\text{horizontal}} \times v = 50 \times \cos(10^\circ) \times 5.55 \approx 273 \, \text{W} \)
Hence, the greatest power exerted by the force is approximately \(273 \, \text{W}\).
(iv) – Finding the time it takes for the block to come to rest:
You’ve used the kinematic equation to determine the time for the block to come to rest. This involved equating the forces acting on the block to the motion equation. The final equation relates time to the velocity and acceleration:
\( 0 = 5.55 – 0.102t \)
\( 0.102t = 5.55 \)
\( t \approx \frac{5.55}{0.102} \approx 54.4 \, \text{s} \)
Thus, the time it takes for the block to come to rest is approximately \(54.4 \, \text{s}\).
Question
A cyclist has mass 85 kg and rides a bicycle of mass 20 kg. The cyclist rides along a horizontal road against a total resistance force of 40 N. Find the total work done by the cyclist in increasing his speed from 5 m s−1 to 10 m s−1 while travelling a distance of 50 m.
▶️Answer/Explanation
The total work done on the cyclist-bicycle system can be calculated using the work-energy principle. The work done will be equal to the change in kinetic energy of the system.
Given:
– Initial speed, \(u = 5 \, \text{m/s}\)
– Final speed, \(v = 10 \, \text{m/s}\)
– Distance traveled, \(d = 50 \, \text{m}\)
– Total resistance force, \(F_{\text{resistance}} = 40 \, \text{N}\)
– Mass of cyclist, \(m_{\text{cyclist}} = 85 \, \text{kg}\)
– Mass of bicycle, \(m_{\text{bicycle}} = 20 \, \text{kg}\)
First, calculate the initial and final kinetic energy of the system using the formula \(KE = \frac{1}{2}mv^2\):
Initial kinetic energy:
\( KE_{\text{initial}} = \frac{1}{2}(m_{\text{cyclist}} + m_{\text{bicycle}}) \times u^2 \)
\( KE_{\text{initial}} = \frac{1}{2}(85 + 20) \times 5^2 \)
\( KE_{\text{initial}} = \frac{1}{2} \times 105 \times 25 = 1312.5 \, \text{J} \)
Final kinetic energy:
\( KE_{\text{final}} = \frac{1}{2}(m_{\text{cyclist}} + m_{\text{bicycle}}) \times v^2 \)
\( KE_{\text{final}} = \frac{1}{2}(85 + 20) \times 10^2 \)
\( KE_{\text{final}} = \frac{1}{2} \times 105 \times 100 = 5250 \, \text{J} \)
\( \text{Work done} = KE_{\text{final}} – KE_{\text{initial}} \)
\( \text{Work done} = 5250 – 1312.5 = 3937.5 \, \text{J} \)
Work done against resistance\(=50\times 40\)
Work done against resistance=2000
Total Work done=3937.5+2000
Total Work done=5937.5J
Therefore, the total work done by the cyclist in increasing the speed from \(5 \, \text{m/s}\) to \(10 \, \text{m/s}\) while traveling a distance of \(50 \, \text{m}\) against a total resistance force of \(40 \, \text{N}\) is \(5937.5 \, \text{J}\).
Question
A load of mass 160 kg is pulled vertically upwards, from rest at a fixed point O on the ground, using a winding drum. The load passes through a point A, 20 m above O, with a speed of \(1.25 m s^{−1}\)(see diagram). Find, for the motion from O to A,
(i) the gain in the potential energy of the load,
(ii) the gain in the kinetic energy of the load. The power output of the winding drum is constant while the load is in motion.
(iii) Given that the work done against the resistance to motion from O to A is 20 kJ and that the time taken for the load to travel from O to A is 41.7 s, find the power output of the winding drum.
▶️Answer/Explanation
(i) PE gain is 32,000 J
(ii) \(KE \, \text{gain} = \frac{1}{2} \times 160 \times 1.25^2\)
Let’s verify the calculations:
\( KE \, \text{gain} = \frac{1}{2} \times 160 \times 1.25^2 = \frac{1}{2} \times 160 \times 1.5625 = 125 \, \text{J}\)
The calculations provided indeed confirm a kinetic energy gain of 125 J.
(iii) Work done by the drum \(= 32,000 \, \text{J} + 125 \, \text{J} + 20,000 \, \text{J} = 52,125 \, \text{J}\)
The time taken \(= 41.7 \, \text{s}\)
The power output of the winding drum can be calculated as:
\( \text{Power} = \frac{\text{Work}}{\text{Time}} = \frac{52,125}{41.7} \approx 1250 \, \text{W}\)
Hence, based on the recalculated values and the provided calculations, the power output of the winding drum is indeed approximately \(1250 \, \text{W}\).