A block of mass 5 kg starting from rest pulled up on a smooth incline plane making an angle of 30◦ with horizontal with an affective acceleration of 1 ms−2. The power delivered by the pulling force at t = 10 s from the starts is W. [use g=10 ms−2] (Calculate the nearest integer value)
A block of mass 5 kg starting from rest pulled up on a smooth incline plane making an angle of 30◦ with horizontal with an affective acceleration of 1 ms−2. The power delivered by the pulling force at t = 10 s from the starts is W. [use g=10 ms−2] (Calculate the nearest integer value)
Solution and Explanation
Given:
- Mass (\( m \)) = \( 5 \, \text{kg} \)
- Angle of incline (\( \theta \)) = \( 30^\circ \)
- Acceleration (\( a \)) = \( 1 \, \text{m/s}^2 \)
- Time (\( t \)) = \( 10 \, \text{s} \)
- Gravitational acceleration (\( g \)) = \( 10 \, \text{m/s}^2 \)
- Initial velocity (\( u \)) = \( 0 \, \text{m/s} \)
Step 1: Resolve Forces and Find Pulling Force (\( F \))
The forces acting along the incline include:
- Pulling force (\( F \)) acting upwards along the incline.
- The component of weight (\( mg \sin \theta \)) acting downwards along the incline.
- The force causing acceleration (\( ma \)) acting upwards.
Applying Newton’s second law along the incline:
\[ F - m g \sin \theta = m a. \]
Substitute the given values (\( m = 5 \, \text{kg}, g = 10 \, \text{m/s}^2, \sin 30^\circ = 0.5, a = 1 \, \text{m/s}^2 \)):
\[ F - 5 \cdot 10 \cdot 0.5 = 5 \cdot 1. \]
Simplify:
\[ F - 25 = 5 \implies F = 30 \, \text{N}. \]
Step 2: Calculate the Final Velocity (\( v \))
Using the first equation of motion:
\[ v = u + a t. \]
Substitute \( u = 0, a = 1 \, \text{m/s}^2, t = 10 \, \text{s} \):
\[ v = 0 + 1 \cdot 10 = 10 \, \text{m/s}. \]
Step 3: Calculate the Power (\( P \))
Power is the rate at which work is done, given by:
\[ P = F v. \]
Substitute \( F = 30 \, \text{N} \) and \( v = 10 \, \text{m/s} \):
\[ P = 30 \cdot 10 = 300 \, \text{W}. \]
Final Answer:
The power delivered by the pulling force is \( 300 \, \text{W} \).
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Concepts Used:
Work, Energy and Power
Work:
- Work is correlated to force and the displacement over which it acts. When an object is replaced parallel to the force's line of action, it is thought to be doing work. It is a force-driven action that includes movement in the force's direction.
- The work done by the force is described to be the product of the elements of the force in the direction of the displacement and the magnitude of this displacement.
Energy:
- A body's energy is its potential to do tasks. Anything that has the capability to work is said to have energy. The unit of energy is the same as the unit of work, i.e., the Joule.
- There are two types of mechanical energy such as; Kinetic and potential energy.
Read More: Work and Energy
Power:
- Power is the rate at which energy is transferred, conveyed, or converted or the rate of doing work. Technologically, it is the amount of work done per unit of time. The SI unit of power is Watt (W) which is joules per second (J/s). Sometimes the power of motor vehicles and other machines is demonstrated in terms of Horsepower (hp), which is roughly equal to 745.7 watts.
- Power is a scalar quantity, which gives us a quantity or amount of energy consumed per unit of time but with no manifestation of direction.