($g = 10 \text{ m/s}^2$).
Force Calculation for Zero Acceleration on Inclined Plane
To find the force required for the block to move down the inclined plane without acceleration ($a=0$), we analyze the forces acting parallel to the plane and set the net force to zero.
Forces Acting Parallel to the Incline
- Component of Gravity: Acts downwards. Calculated as $W_\parallel = mg \sin\theta$.
- Kinetic Friction Force: Acts upwards (opposing downward motion). Calculated as $f_k = \mu_k N$, where $N = mg \cos\theta$.
- Applied Force: External force $F$, acting parallel to the incline.
Step-by-Step Calculation
- Given Values:
- Mass, $m = 5 \text{ kg}$
- Angle, $\theta = 30^\circ$
- Friction coefficient, $\mu_k = \frac{\sqrt{3}}{2}$
- Gravity, $g = 10 \text{ m/s}^2$
- Calculate Downward Gravity Component:
$W_\parallel = mg \sin\theta = (5 \text{ kg})(10 \text{ m/s}^2) \sin(30^\circ) = 50 \times \frac{1}{2} = 25 \text{ N}$
- Calculate Normal Force and Friction:
Normal Force, $N = mg \cos\theta = (5 \text{ kg})(10 \text{ m/s}^2) \cos(30^\circ) = 50 \times \frac{\sqrt{3}}{2} = 25\sqrt{3} \text{ N}$
Friction Force, $f_k = \mu_k N = \left(\frac{\sqrt{3}}{2}\right) (25\sqrt{3} \text{ N}) = \frac{25 \times 3}{2} = \frac{75}{2} = 37.5 \text{ N}$
- Net Force Equation for Zero Acceleration:
Assuming the applied force $F$ acts downwards along the incline, the net force $F_{net}$ along the incline is the sum of forces acting downwards minus forces acting upwards.
$F_{net} = W_\parallel + F - f_k$
For zero acceleration, $F_{net} = 0$:
$25 \text{ N} + F - 37.5 \text{ N} = 0$
- Solve for Applied Force $F$:
$F = 37.5 \text{ N} - 25 \text{ N}$
$F = 12.5 \text{ N}$
The required applied force is $12.5 \text{ N}$ downwards along the incline.
Similar Questions
- When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes $15$ divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is ________mm.
(Least count of Vernier calliper = $0.1 \text{ mm}$) - Water drops fall from a tap on the floor, $5 \text{ m}$ below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ________m.
($g = 10 \text{ m/s}^2$) - Two wires $A$ and $B$ made of different materials of lengths $6.0 \text{ cm}$ and $5.4 \text{ cm}$, respectively and area of cross sections $3.0 \times 10^{-5} \text{ m}^2$ and $4.5 \times 10^{-5} \text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is________.
Two circular discs of radius each $10 \text{ cm}$ are joined at their centres by a rod of length $30 \text{ cm}$ and mass $600 \text{ gm}$ as shown in figure.
If the mass of each disc is $600 \text{ gm}$ and applied torque between two discs is $43 \times 10^5 \text{ dyne.cm}$, the angular acceleration of the discs about the given axis $AB$ is________$\text{rad/s}^2$.
- A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = -kv$, where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity ($v$) versus time ($t$)?
- A solid sphere of radius $10 \text{ cm}$ is rotating about an axis which is at a distance $15 \text{ cm}$ from its centre. The radius of gyration about this axis is $\sqrt{n}\text{ cm}$. The value of $n$ is
- A projectile is thrown upward at an angle $60^\circ$ with the horizontal. The speed of the projectile is 20 m/s when its direction of motion is $45^\circ$ with the horizontal. The initial speed of the projectile is _________ m/s.
Net gravitational force at the center of a square is found to be $F_1$ when four particles having mass $M, 2M, 3M$ and $4M$ are placed at the four corners of the square as shown in figure and it is $F_2$ when the positions of $3M$ and $4M$ are interchanged. The ratio $\frac{F_1}{F_2}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is _________.
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
- Match the LIST-I with LIST-II
List-I List-II A. Spring constant I. $[\text{M L}^2 \text{ T}^{-2} \text{ K}^{-1}]$ B. Thermal conductivity II. $[\text{M L}^0 \text{ T}^{-2}]$ C. Boltzmann constant III. $[\text{M L}^2 \text{ T}^{-3} \text{ A}^{-2}]$ D. Inductive reactance IV. $[\text{M L T}^{-3} \text{ K}^{-1}]$
Choose the correct answer from the options given below:
Important Questions from Mechanics
- When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes $15$ divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is ________mm.
(Least count of Vernier calliper = $0.1 \text{ mm}$) - Water drops fall from a tap on the floor, $5 \text{ m}$ below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is ________m.
($g = 10 \text{ m/s}^2$) - Two wires $A$ and $B$ made of different materials of lengths $6.0 \text{ cm}$ and $5.4 \text{ cm}$, respectively and area of cross sections $3.0 \times 10^{-5} \text{ m}^2$ and $4.5 \times 10^{-5} \text{ m}^2$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x : 3$. The value of $x$ is________.
Two circular discs of radius each $10 \text{ cm}$ are joined at their centres by a rod of length $30 \text{ cm}$ and mass $600 \text{ gm}$ as shown in figure.
If the mass of each disc is $600 \text{ gm}$ and applied torque between two discs is $43 \times 10^5 \text{ dyne.cm}$, the angular acceleration of the discs about the given axis $AB$ is________$\text{rad/s}^2$.- A particle of mass $m$ falls from rest through a resistive medium having resistive force, $F = -kv$, where $v$ is the velocity of the particle and $k$ is a constant. Which of the following graphs represents velocity ($v$) versus time ($t$)?
